Commonly, isolation core (or ab) exercises, such as crunches are performed. I feel that these exercises are not an efficient means of core training for many reasons.
First of all, the most effective training program is one that is tailored to an individuals current condition and their specific goals. So, there may be a situation when one of these exercises may be used effectively. With that said, a crunch or a variation has little transfer to other exercises, movements, or activities of daily living. The specific neuromuscular recruitment patterns, which you are developing with crunches, differs significantly from the neuromuscular patterns you need to effectively perform other exercises or activities. Add that to the fact that crunches reinforce bad posture, repeatedly flex (bend) the spine ( which research shows is very bad for your the discs in your spine), and exacerbate muscle imbalances (ideally you want to have greater recruitment of the Internal and External Obliques than the Rectus Abdominis).
'Core' training should emphasis stabilizing the pelvis, spine, and shoulder blades so the limb muscles have a strong foundation to pull against. The 'Core' muscles also function to resist spinal bending or twisting. The 'core' muscles essentially 'tie' the arms and legs together (think of a golf swing or pitching a baseball) and help transfer forces effeciently from the legs to arms or vise versa.
So, a better 'core' exercise is the Plank.
You can see how the arms are 'tied' to the legs. Also, additional muscles, working together, stabilize the shoulders, and hips. I use the Plank with many beginners to teach them how to 'brace' their torso.
As, a client adapts to the Plank (they can hold the proper position for more than 0:45), then it is time to progress the 'core' exercises. The Arm Bars are much more challenging, as additional shoulder and hip stabilization is required.
While the Plank is a good exercise to train basic stabilization, I feel it's transfer to other, upright exercises and activity is minimal. Exercises that involve arm and/or leg movment or variations of the main exercises (squat, deadlifts, presses, etc) thay you are trying to improve are necessary. This is where your Turkish Get-Ups, Suitcase Deadlifts, and Single-Arm Push-Ups come into play.
Finally, the basic, compound, free-weight exercises require tremendous 'core' stability. Simply performing these exercises with a moderate resistance will enhance 'core' stability, as evident by improvements in your one-repetition maximum. 'Core' instability and fatigue are often the limiting factor for these exercises.
Research shows that 'core' activation is significantly higher during squata and deadlifts than during so-called 'core'-specific exercises. Just don't try these exercises on an unstable surface- that is just defeating the purpose! The higher activation of the 'core' muscles is necessary to counter the powerful pull of the limb muscles.
Hopefully, these examples will help you better understand the role of the 'core' muscles and how performing many of the popular 'core' exercises you see in fitness magazines and at the gym are ineffective and a waste of your time.