Ok, you both made me laugh a little bit.
Basically, I don't really think rings or gymnastics stuff is going to turn you into a ninja. (You'd probably be better off with parkour/freerunning or something like that.) That's not to say it's not awesome in its own way, but I don't think it's some holy grail or anything.
There really is almost nothing out there about combining strength-based work with rings. I've dug through a number of places, but can find very very little. Lots of crossfit things, and a fair amount of gymnast things, but all that is bodyweight and for either endurance/reps or skill work.
Short story is, I moved, my rack went into storage, and all I had to work with was a barbell. So for several months, I was just doing an overhead press variation, and a deadlift variation. I then got some rings for Christmas, and after getting past the initial excitement of wanting to be a ring gymnast and do all this awesome stuff, I added in ring dips, ring pullups and hangs for time. Recently, I also added some wrist roller work, with a fat bar wrist roller suspended in the rings.
1. deadlift variation (mid-shin mat pulls with chains), trained in the style from the book Power to the People. For a given day, do a max set of 5, drop 10% (~30lbs), do another set of 5. Next session, add ~5% (15lbs), do the same. Keep adding weight each session to hit a new peak, then drop to 70% and restart a new cycle.
This schedule has gotten a bit messed up, as I've been changing my overhead programming and training in a different gym twice a week. I may change the routine up, not the lift.
The mat pulls seem to work better than the deadlift, and the chains keep the weight heavy through the whole ROM. Otherwise, the start of the lift is heavy but the finish is easy. Goal here is to maximize muscle stress through the whole movement. These are done touch and go, very controlled.
overhead press variation (strict press from rack, or clean and press... varies). I'm using the routine from the book "Hepburn's Law". Basically, 8 sets of 2, working to 8 sets of 3. Then 6 sets in the 4-6 rep range. Then a final "pump" set. The book explains it well, but this is 2x a week.
ring pullups. Grip starts out pronated, moves to neutral as I finish out the rep. These are done using the 3RM schedule from Pavel's "Fighter Pullup" routine. See the "3RM Russian Pullup Program" section here: https://www.T-Nation.com/workouts/how-to-increase-your-pull-up-power
I do that, then add 5 lbs and start over.
ring dips. Same thing.
fat-bar supported wrist roller. I just recently added them in, but this is to increase my grip strength, and to improve my wrist strength. Wrist strength has been an issue for pressing, so I'm trying to fix it.
That's pretty much it. A little bit of arm stuff here and there (one-arm pushdowns), to try and even a few things out.
The dips, pullups and presses seem to be making a lot of changes in my upper body. The dips especially, since on the rings I seem to hit everything from my arms to my lats to my chest, as well as a whole bunch of "stabilizer" muscles around the shoulder joints (teres, supraspinatus, rhomboids, mid and low traps). On parallel bars, I don't get anywhere near the muscle involvement as on the rings.
The mat pulls with chains basically hit everything else on the midline of the body, from the neck to the calves. There's actually a fair amount of quad and abductor involvement too, for being primarily a posterior-chain movement.
I like the combination of everything I'm doing right now; it seems to hit what I want, and make me better where I want.