Oh that's broad. I have a LOT of bits for a LOT of different things.
Right now since most of my horses will do both gaming and performance events, they will have a gaming bridle and a show bridle.
Almost all my schooling at home is done in a plain myler comfort snaffle, smooth or twisted depending on the horse.
I'll go with just my two main kiddos right now.
Selena does a long shank Billy Allen for reining, the cowhorse, equitation, etc. She is ridiculously soft in the face so doesn't need much mouthpiece, I choose the long shank on her though just as an "in case measure." Then for the gaming, barrels, drill, any "go hard" events she gets a short shank smooth jr. cowhorse dogbone with copper rollers. It's extremely light. It's enough bit that I can take a hold of her when she gets strong but not enough that if I snatch her on accident she's going to get slowed up and bound by my hand. That teensy bit of gag just says "come here" when I need it to.
Ruger right now is still mainly in a snaffle. He is four and coming along very well, but every now and then I'll put him in my o ring dogbone twist combo. Mainly because he's lifty and I can shut down that front end pretty fast with the gag and nose pressure if he comes up too far, but admittedly it's also because he inhales shanks and reins if he can...Even if the shank is long he will suck it in. Put him in Selena's cowhorse when I first started getting him up in the bridle and almost instantly he had the shank and half my reins in his mouth and there was nothing I could do about it.
However, these bits are relative. I like to use a lot of Myler bits to show in, as well as the billy allens. I'm sold on those as far as reining and other performance events goes. The gaming is a whole 'nother story though. I have a LOT of different options there.
Pssh.I didn't pick up the wrong lead
It's called a counter canter...
...A very advanced maneuver.