[Madmick]

INTRODUCTION

For this sticky, I tried to restrict the list to exercises that are more intensive for the core than other muscle groups. Obviously, the deadlift is one of the most potent core exercises, period, but it is more intensive for the posterior chain as a whole than simply "the core." The same goes for squats. Nevertheless, if you're doing hyperextensions and not deadlifts or squats, you need to start from scratch.

To get a better understanding of the anatomy of "the core", and as a primer for developing your routine, I recommend you read this article by Mike Robertson over at T-Nation (or better yet, buy the best S&C book on the market for fighters: "Inifinite Intensity" by Ross Enamait). Several of the exercises listed below are covered in this article:
21st Century Core Training by Mike Robertson

The core is worked in three different planes: 1) Frontal, 2) Sagittal, and 3) Transverse. In simple terms: Frontal is bending forward and backward; Sagittal is bending sideways; and Transverse is rotating/twisting. At Urban's suggestion, I categorized them below according to those movements types. Strength in all three planes are essential for a martial artist, and although you cannot neglect any, for the most part (and especially for knockout power and throwing power) the most important is to possess power in the transverse plane.

I have added two other sections: Static/Stabilization and Dynamic.

Static/Stabilization is for exercises that require mostly isometric strength in the core. Although some of these exercises require movement, they are mostly intensive in an isometric fashion. Take, for instance, the classic "Bicycle" exercise where you lie on your back and pedal your feet in the air as if you were riding a bike. Although you are moving your limbs, anyone who has done this exercise knows that the burn is mostly static in the upper and lower abdominals.

Dynamic exercises are just that: exercises for the core performed dynamically. Since any movement can be performed dynamically, I've tried to restrict this list to exercises that don't have a corresponding slow tempo, but are intended to be performed exclusively dynamically, so that those unfamiliar with these exercises won't make the mistake of performing them in the standard slow tempo. For example: Sledgehammer Swings. I'm sure you'd get some benefit from hitting a tire with a sledgehammer in a slow, controlled fashion, but it's mostly a waste of time to do this. This may seem obvious, but with exercises like the Full Contact Twist, it is less so. Additionally, since exercises in this list may work in any of the three planes, I have listed its plane in brackets next to it.

Note: if you have a link with a better video/demonstration of an exercise, please suggest it. If you think a linked demonstration has improper form, please voice that. If there is any way you feel that this sticky can be improved, just post.

Keith Wassung's Big 4 for Core
1) Overhead Squats
2) Decline Sit-Ups w/Controlled Cadence
3) Waiter Walks
4) Saxon Side Bends



THE CATALOGUE

Abdominal Flexion (Frontal Plane)
Ab Wheel Roll-Outs
The linked video is of standing ab-wheel rollouts: an advanced variation. Start will rollouts from your knees. You can also do this with a simple barbell (given that your plates are non-hexagonal). For more information on how to progress from knee rollouts to standing rollouts (and eventually one-arm standing rollouts), I recommend the book "Never Gymless" by Ross Enamait.
Dragon Flags
An advanced exercise popularized by Pavel Tsatsouline. A beginning variation would be the reverse crunch (where you simply curl up the knees to touch your elbows).
Turkish Get-Ups
The linked video is just one method of the Get-Up. The only rule to the TGU: the extended arm with weight cannot be bent while getting up.
V-Ups
The linked video (from ExRx) is a basic V-Up; not a weighted a V-Up. I don't know why it says that. To weight the V-Up, hold a heavy medicine ball in your hands. Martin Rooney calls this exercise the Pike-Up, and wrote in his manual "Training for Warriors" that a pike-up with a 50-pound medicine ball is one of his favorite core exercises.
Sit-Ups/Weighted Sit-Ups/Decline Sit-Ups
I'm assuming everyone knows how to do a sit-up, so I linked a lying weighted sit-up. That is one way to do it. You may also hold the weight (medicine ball, plate, DB, whatever) behind your head as long as you don't crank your neck when you sit-up. The second video is the more advanced Decline Sit-Up. With the decline sit-ups, I recommend holding the weight on your chest. If you want to increase the difficulty, either increase the weight or the grade of incline.
Swiss Ball Crunches (Variation #2)
The swiss ball crunch is a variation of the normal sit-up/crunch that requires more stabilization. Some simple tricks to increase the difficulty by a small margin is to bring your feet close together and push your body back so that your center of gravity is further from your feet. #2 is a weighted variation of the exercise called the Reaching Swiss Ball Crunch. The pictures I have linked show the demonstrator keeping the medicine ball overhead throughout the crunch. For a more advanced variation, bring the ball back as you would in the V-Up so that your arms are parallel to the floor.
Standing/Kneeling Ab Pull-Downs
The linked picture sequence is of a standing ab pull-down. The kneeling variation is performed the same way, but on the knees.
Captains Chair
The beginner's variation is to just raise the knees; the advanced is to raise the legs while straight. I personally recommend a slow tempo (3-2-1 or slower) while doing this exercise. It's useful for the hip flexors. Another variation is to swing the legs up, held straight, one by one.
Hanging Knee/Leg Raises
The more advanced variation of the knee/leg raises on the Captain's Chair, here you simply hang from a bar while raising the legs. The video I've provided is of the more advanced straight leg raise; an easier variation, as with the Captain's Chair, is to just raise the knees with the lower legs dangling.
Leg Raises
What this guy knows that the rest of us don't is that this exercise is more effective when you rock the Ashlee Simpson.
Floor Wipers
Mike Martial contributed this advanced variation of the leg raise. There's a bit of transverse (and stabilizing) work in this one, but it looks most intensive for abdominal flexion. Plus it honors the movie 300. Byyaaahhh!!!
Jack-Knifes/Butt Rows
An exercise everyone in martial arts should be familiar with. The Butt Row variation involves keeping the torso erect while rowing the legs in and out. A variation Rooney suggested in Training for Warriors is to add cable resistance by hooking the handle over the feet (I'd wear shoes with this variation).
Chinnies
Sorry this is the best I could find, but simply repeat this motion back and forth (touching elbows to knees) again and again without letting your feet or head touch the floor. Although you do involve your obliques to an extent, without the hips coming off the floor, I don't feel there's enough rotation for this to be classified in the Transverse Plane. So it goes here.

Abdominal Extension (Frontal Plane)
Ab Wheel Roll-Outs
Yes, you go forward and backward, so I've listed them in both Flexion and Extension. Once again, the linked video is of standing ab-wheel rollouts: an advanced variation. Start will rollouts from your knees. You can also do this with a simple barbell (given that your plates are non-hexagonal). For more information on how to progress from knee rollouts to standing rollouts (and eventually one-arm standing rollouts), I recommend the book "Never Gymless" by Ross Enamait.
Overhead Squats
Perhaps the most challenging core exercise there is. I previously categorized this exercise in the "Static/Stabilization" section (since your spend most of your energy stabilizing the core), but decided to move it here with the more burly exercises.
Neider Press
Although I couldn't find a picture or video of the Neider Press, it's easy to describe: simply perform a bench press in a standing position. You'll have to maintain tension in posterior chain, especially your lower back, to maintain balance. Another variation is to perform it at a diagonal (like an incline bench).
Suitcase Deadlifts
It's the fourth exercise Mahler lists. This can be done with a Kettlebell, as shown here, or a DB, or a Barbell. You can also do both sides simultaneously.
GHR (Variations)
The ultimate auxiliary exercise to the deadlift and the squat. #2 shows several variations. You might have to improvise if your gym doesn't stock a GHR machine (it likely doesn't).
Hyperextensions/ Zercher Hyperextension
You can also do this exercise lying on your stomach on a swiss ball.
Reverse Hyperextension
I believe this is also called the Reverse GHR, but I'm not sure.
Skydivers/Supermans
Simple bodyweight exercises that are best utilized for warming up, in a bodyweight circuit (where their low intensity will be challenging in a fatigued state), or as an auxiliary exercise for a weighted movement. An advanced variation we did as swimmers was to reach back and grab the heels with both hands, and raise as much of the body off the floor while maintaining floor contact with the chin. This variation requires flexibility as much as strength.

Lateral Deviation (Sagittal Plane)
Saxon Side Bends
Perhaps the best exercise in the sagittal plane.
Windmills
Many prefer to do this with a Kettlebell (as shown here), but you can do it with a DB just fine.
Side Presses/Bent Presses
The Side Press and Bent Press are two different (but very similar lifts). The best illustration and explication of their differences I've found is here on Mahler's page. They're about a 1/3 of the way down. Go ahead and take the time to check out the other lifts he demonstrates, too (including the Windmill).
Side Deadlifts
You can use a barbell, preacher bar (as shown here), dumbbell, girlfriend...whatever.
DB Side Bend
You can load yourself unilaterally or bilaterally.
Lateral Bridges
If you hold it at the top, it's a "Side Plank" (see "Plank" in the Static/Stabilization section below). With motion, it becomes the Lateral Bridge.
Side Crunch
Personally, I think this exercise sucks, but it's an old standard.
Pendulums
The Pendulum is an exercise where you lay on your back with your hands pointed down at a 45 degree angle away from your body. Raise your legs off the floor. Now, keeping your legs at the same height, swing them from one side of your body to the other. This is a great exercise for a bodyweight circuit.
Standing Cable, Band or Machine Lever Abductors/Adductors
The bodyweight version of the adductor exercise is the lying side leg lift (lie on your side, lift your top leg up).

Rotational (Transverse Plane)
Twisting Deadlifts
Unfortunately, I couldn't find any illustration online of this exercise (you'll find one in Enamait's "Infinite Intensity"). It's fairly easy to describe, though. You begin with the weight (I use a dumbbell) on one side of your body- as though you were doing a side deadlift. But instead of picking the weight up with the hand on the same side, you twist and pick up it up with the hand from the opposite side. You lift the weight in a twisting motion back along that plane to where the lifting hand began, then lower it straight down to the side of your foot. Now the weight is in place so that you can repeat to the other side.
Decline Russian Twists (Swiss Ball Variation)
Although the Russian Twist has many variations (including the Swiss Ball one I've shown), easily the best is the Decline Russian Twist (w/added resistance). Some guys like to do Russian Twists with a partner where they sit back-to-back and hand a medicine ball off- back and forth- from side to side.
Cable (w/Physioball) Twists
Okay, this guy has terrible technique (as well as terrible taste in music), but I just wanted a visual. The mistake he's making is that he isn't turning his head with his body, and he's not really keeping his arms straight and rotating from the hips (he's using his shoulders too much). Anyway, this is the High Cable variation (use the highest setting possible). The Standard Variation is to pull the cable from shoulder height, and the Low Cable variation is to pull from the feet. For added difficulty, cradle a physioball (or medicine ball) in your arms. This adds weight, but more importantly, it forces good form and keeps you from involving your arms like this douche.
Hip Swings
A great bodyweight exercise in the transverse plane. This variation shows the guy holding a swiss ball between his feet, but that isn't necessary.

Static/Stabilization
Waiter Walks
Lift something over your head. Now walk around. Walk over things (like benches). Walk up a flight of stairs. You can do this exercise unilaterally (with one arm weighted) or bilaterally (with both arms weighted).
Vacuum/Hoover
I know this exercise by both names ("Hoover" is the old strongman title, I think). This guy doesn't seem to be doing very much. I remember when I was a kid I swear I could make my belly button touch my spine. Try to hold for 30 seconds.
Plank
A basic position. It's not very tiring in itself, so usually I incorporate it as active recovery in a bodyweight circuit (for instance, fatigue yourself with burpees and pushups, then see how easy it is to hold this position).
Leg Holds
Lie on your back with your body straight and your arms by your side. Raise your legs an inch off the ground and hold. A classic variation is to adduct your legs varying distances (as in the Chinese splits) and hold.
Spears
A more advanced variation of the Leg Hold, and a great transition position between other bodyweight static exercises in an ab circuit. Lie on your back with your body held straight and your arms overhead (your hands overlapping in a streamlined position). Raise your legs and head an inch off the ground. Hold. Hold your legs at a 45 degree angle from the floor for maximum difficulty.
Flutter Kicks
From the Leg Hold or Spear position, now kick your feet like you would if you were swimming freestyle (keeping the legs straight, up and down alternately, varying the distance your kick). Do not let your feet touch the ground.
Dolphin Kicks
From the Leg Hold or Spear position, now kick your feet like you would if you were swimming butterfly (keeping the feet together, legs straight, raising and lowering them several inches). Do not let your feet touch the ground. The bigger you motion the more challenging it is.
Scissors
From the Leg Hold or Spear Position, now crisscross your feet horizontally over and under one another. Once again, do not let them touch the ground.
Bicycles
Pedal in the normal direction (as shown here), but also reverse that direction.

Dynamic
Sit-Ups w/Overhead Medicine Ball Throws [Frontal]
If you don't have a partner, find a wall, and throw it off that.
Medicine Ball Slams [Frontal]
The second portion of this video is the slam (with the medicine ball). I like the swing-slam combo. You can use the medicine ball for the swing portion if you prefer.
Overhead Medicine Ball Throws [Frontal]
The wall can work with this exercise, too.
Backwards Overhead Medicine Ball Throws (aka BOMBs) [Frontal]
I couldn't find a vid, but you begin by standing, holding the ball with both hands between your legs, and explosively throw the ball backwards over your head. An inferior variation is to throw the ball underhanded back and forth between yourself and a partner.
Double Leg Throws (Partner-Assisted) [Frontal or Transverse]
Don't let your feet touch the ground. Tell your partner to throw your legs hard, and to wear pants.
Alligators [Sagittal]
Grab two frisbees, and turn them upside down. Get in the pushup position. Lower yourself until you're an inch off the ground. Place each of your feet in the frisbees. Now, staying as low to the ground as possible, crawl like an alligator as quickly across the floor as you can (your feet will slid in the frisbees).
Full Contact Twists [Transverse]
Generally considered one of the best exercises for increasing knockout power.
Power Iron Coil
I've seen Fedor do this in a training vid, but I can't find it.
Sledgehammer Swings [Frontal or Transverse]
Here is a link to Ross's article (including video demonstrations). Hitting a tire on the ground is mostly intensive in the frontal plane (although there is transverse work). If you swing at a target to the side, it will be solely in the transverse plane. Cutting a tree with an axe works great.
Woodchoppers [Transverse]
Pretty much the same thing as a sledgehammer swing. The linked video shows a special implement that is a rope with medicine ball attached. You can use a plain old medicine ball, though, or even a cable (like the High Cable Twist shown above, but performed dynamically...just anything with a pulley and weight).
Medicine Ball Side Throws [Transverse]
Boxer's Seated DB Twist [Transverse]
This is another exercise I saw in Rooney's book. Sit on the floor. Grab a dumbbell and hold it in front of you. Raise your legs. Now twist as quickly as possible from side to side. I've had difficulty keeping my legs forward. Thought I'd list it anyway. You can also do the Decline Russian Twist dynamically. More or less, it's the same thing.