The handstand on the ground is fairly tough but easily doable with a little practice. The handstand on rings is much, much tougher.
Here we have Jim Bathurst on the rings working on his handstands.
What makes ring handstands harder to do? You can’t use your wrists at all in order to balance like you can on the floor or parallel bars. Instead all the balance must be done from the shoulders and by bending the arms as you can see Jim do quite a bit.
The truth is ideally you don’t want to bend your arms just like in a normal handstand, but when you’re first starting out it may happen. Try to avoid it though as it can set in bad habits.
The easy way to do a handstand on the rings is to wrap your legs around the straps. This is comparable to doing a handstand against the wall. Of course the freestanding ring handstand and that against the straps is like a freestanding handstand versus one against the wall, just the rings both are a jump in difficulty.
Still if you work on handstands on the rings your balance will grow that much stronger.
Some other lead up stunts to it would include shoulder stands on the rings and many other skills.