The following exercises are my favourite gymnastic ring exercises. They are demanding and you must have a good foundation of strength before attempting these. If you have a TRX suspension system some of these can be performed using them.
Muscle ups – targets lats, chest, triceps. This exercise is possibly one of the most difficult to perform. It is essentially a pull-up followed by a dip.
Dips or L dips – targets chest and triceps. Dip down until your elbows are 90 degrees. Extend your elbows. For added difficulty and to challenge abs do a dip and raise your legs at the top of the movement.
Archer Push-ups – targets chest. Archer push-ups are essentially a push-up on rings with a fly.
Archer Pull-ups - targets lats and biceps. Archer pull-ups are a good way to progress to one arm pull-ups.
Tricep Extensions – targets triceps. Set up the rings to around hip height. Grab onto the rings with elbows at ears and perform an overhead extension using your bodyweight. This is a very difficult exercise for the triceps. Beginners may want to avoid this one.
Inverted row – targets lats. for those who cannot perform a full body pull-up, rings are a great progression tool. Set the rings to about hip height. Try to row your body up while keeping legs straight. If this is too difficult, bend knees.
Pistol Squats – targets quads, glutes. Rings are a great tool to help you to progress to bodyweight pistol squats. Use one ring to help you lower down and rise up out of the squat. Pull as much as you need until your legs are strong enough to do standard pisol squats.
Here are some really good gym ring and TRX training reference materials: