Shoulder yoga includes any yoga pose where the shoulders are used to support some or all of the weight of the body. It also includes shoulder stretches and yoga arm positions that stretch or exercise the shoulders.
One of the best ways to begin a yoga shoulder program is to start with shoulder awareness exercises, particularly exercises that focus on learning to feel, control and stabilize the shoulder blades. These exercises can start with fairly relaxed movements of the shoulder blades without the arms bearing weight and then moving towards poses where the arms are bearing weight.
As part of the process of learning to feel the shoulder blades it can be helpful to learn the reference points or landmarks of the shoulder-blades, particularly ones that you can feel or palpitate with the other hand.
An important aspect of developing shoulder control can be that of learning to feel the ribcage and to separate movements of the ribcage from those of the shoulders and vice versa. This isn't the end goal but rather an intermediate step so that you have control and awareness of both body parts. They can then be used more effectively together.
One of my favorite exercises to practice isolating and then integrating shoulder and ribcage movements is table top yoga pose.
A simple pose where the shoulders can be exercised while laying on the back is bridge yoga pose. In this pose the shoulders can be pressed down into the earth to help lift the ribcage up away from the floor.
Shoulderstand and plow pose are similiar in this regard. In both poses you can focus on pressing the shoulder down into the floor to help lift the ribcage up off of the floor.
To develop better control of the scapulae I'd suggest practicing this action first with the hands unclasped. Once it is easy to do with hands un clasped, then experiment with clasped variations.
For shoulder flexibility it can be helpful to practice relaxed arm stretches where gravity provides the driving force. The focus in these can be on relaxing the targeted muscles, or on engaging to resist the stretch and then relaxing. Another type of shoulder stretch is to use muscle power, perhaps that of the opposite arm, to drive the stretch.
In all of these poses the elbows are straight and the shoulders can be used to push the ribcage away from the floor.
A slightly more complex yoga pose for the shoulders is chaturanga dandasana. It is more complicated to teach because the elbows are bent.
Yoga inversions are another type of yoga pose where the arms or shoulders bear some or all of the weight of the body.
In headstand the head can bear some body weight with the shoulders and elbows used for stability. In forearm stand and handstand the arms bear all of the weight of the body. In all actions shoulder awareness as well as strength can be useful. Awareness can be used to help do these poses with minimal effort while at the same time maintaining the integrity of the pose.
A further set of exercises to practice for shoulder awareness is arm rotations, particular of the upper arm with respect to the shoulder blade. These can be used as rotator cuff exercises.
Rotated arm positions may be used to help stabilize the shoulder and shoulder blade giving the muscles that attach the shoulder blade to the ribcage and spine a firm foundation from which to act on the ribcage and spine.
A movement practice that can be used to improve shoulder flexibility and control as well as arm rotation is the dance of shiva.
Yoga Shoulder Exercises and Stretches includes a systematic approach to shoulder awareness and flexibility