Relating back to the definition of stability: resistance to change. With this in mind, core stability (maybe more aptly named Lumbar spine stability) means that the lumbar spine should remain the same shape during most exercises. Here is where I would start with training for rehab to performance.
First, can they "BRACE" their abdomen using their core muscles (all the muscles I spoke of earlier in part 2), with emphasis on the abdominal muscles (rectus and transverse abdominis, internal and external obliques. There are a number of methods of teaching this, but the one I found that works best is to get someone to brace as if they were going to get punched in the stomach. Note that you ARE NOT drawing belly button towards spine, this would actually decrease stability.The abdominal brace is best taught with the subject lying on their back. They should be able to brace without moving their back. They should also be able to brace and talk to you while doing it (so they should be able to breathe while bracing). Some people may feel that this is too simple of an exercise and that they don't get anything out of it. If that is the case, you are probably doing it wrong.
Another simple core exercise is bridging. Bridging is a great exercise because it emphasizes stability of the lumbar spine against gravity. The goal of bridging should be to brace abdomen and maintain what I would call a "standing posture" or what your back would look like if you were standing upright. Bridging can be done on elbows in the prone position (or facing the ground), the supine position (facing up) or on sides. It should be tailored to the level of the subject. BRIDGES SHOULD NOT BE HELD TO MAXIMAL TIMES AS THE MUSCLES BECOME ISCHEMIC (no blood in the muscle) THIS CAUSES FATIGUE AND WILL DECREASE PERFORMANCE. I recommend holding no longer than 10 seconds and increasing the number of repetitions you do (ie, instead of holding for 1 minute, do 6 sets holding 10 seconds with a few seconds rest between). 
Another simple core stability exercise is what is known as a "Bird dog" or "Superman" exercise. This is a bit more of a complex exercise as you are on your hands and knees, you are required to brace your abdomen and move the opposite leg and arm (ie, right leg, left arm). With this exercise, the goal is to make sure that the only movement that occurs is from the hip and shoulder/shoulder blade. The more stable you are, the less wobbly you will be.
Things to remember: Core is not an "ON/OFF" switch, you are using it no matter what you are doing (even while you are sitting here reading!) Core is more like a volume that increases with the difficulty of a task. Lifting 10lbs will require a bit of core. Lifting 100lb will require quite a bit more core. Most exercises you do from squatting, deadlifting, pushups and pull ups will require you to challenge core so I recommend staying away from machines. I would consider the most challenging core exercises plyometrics, power cleans and sporting events such as golf, tennis, baseball that require a combination of strength and stability to achieve success.
Please feel free to leave comments, suggestions and questions. Let me know what you think!
