Keg: At least for the ones that are into stallion on stallion.
Breakfalling is a big thing that stuck with me too. More than a decade later I know that I can and will still do them. I'm pretty sure there's more than a few times where I avoided serious injury while snowboarding because of it (and I've heard of enough injuries on the hill during my few year stint that I think it should be a requirement to learn it before starting boarding).
I took Hapkido, which pretty much had the problems d Roy was complaining about for traditional striking martial arts. It was also kind of a jack-of-all trades type martial art, so while we learning grappling and how to get out of them, it was 95% drills. We did do sparring, but it was pretty much a once every six classes type deal. It was my favourite part, though, particularly because it broke up the repetitiveness.
I can't exactly remember what led up to it, but I ended up in a match with my third degree black instructor when I was around the orange level. It was pretty much just him completely shutting down my offense and countering the whole time. I do remember being really happy that I managed to sneak one blow (that would have been entirely inconsequential, but still) in under his guard, though.
Back to relating it to ponies, yes, definitely think mane and tail grabbing would have to be out as far as organized competitions go, especially with how full and flowing most of them are.