In any canon, do we ever see or find out if The Joker is or has trained in any sort of martial art or fighting style?
I've seen him go toe-to-toe with Batman before, I can't remember which animated movie it was but The Joker was armed with a flick knife and put up a good fight. I think they were in an abandoned building...
The following two scenes don't really show off his fighting skills, but they at least demonstrate his quick reflexes and strength across multiple canons. I suppose these attributes could be attributed to his hard life, as opposed to any training.
Generally he seems to be able to take a beating from Batman, but again, his endurance could be down to regular old insanity and not professional training.