Here's a very advanced tutorial on how to Hold a Handstand On Gymnastics Rings.  

I had an audience member ask me how he could learn how to do a handstand on Gymnastics rings which is really stinkin’ difficult because in case you haven’t noticed, rings, unlike bars or the Ground, are not stable.  Which makes it impossible to do a handstand on them... If your handstand isn’t fundamentally correct.  Just as in my how to hold a handstand video, I explained how it all comes down to posture, practice, and some strength.  This is even more important when on the unstable rings.  What if you were trying to stand on rings and balance on your feet???  You better have some solid posture or else you would fall, right?  So what makes standing on your hands on the rings any different?  Nothing.  Except your arms are not as strong as your legs, you can grip the rings better with your hands than your feet... and your face won’t get all red if you are standing on your feet... and you better have a really good handstand to even attempt this...

Once your handstand is competent on the floor, and you can hold it for a while, like more than 30 seconds, then you can try on the rings.  Make sure you have a mat or something underneath you in case you fall over... not like me with a solid concrete floor underneath me.  To start, you can wrap your feet around the rings straps for balance until you get your bearings, then as you become more comfortable, move your feet to just barely touching the inside of the rings straps.  Remember what I said in my how to hold a handstand video about pushing up and getting rid of any shoulder angle?  yah, this is critically important.  You need to extend and push on those rings and make your body as tall as possible.  Also, you’ll want to turn the rings out so your arms don’t brush the straps and try to keep your arms locked out.  In gymnastics, bent arms and touching the straps are a deduction.  Bad.  You can adjust your balance using your wrists, and even your shoulders can planche forward to adjust your balance if needed.  

If you are not there yet and not quite ready for the rings, I recommend practicing handstands against a wall, and even one arm handstands against a wall.  This will help improve your handstand and get you ready for holding it longer on the ground.  Also, handstand pushups and presses are a great thing to practice and will make your handstand better over time.  To help you get rid of that shoulder angle, make sure you regularly stretch your lats and everything else by hanging on a bar.  Also, actively strengthening your scaption muscles is a great idea.  You can do TRX Letter I’s for that - which will also help your reverse hecht for you gymnasts out there.  

Video Link -