I can only speak for agility, but your dog can compete almost anywhere! I run my mix and non-AKC dogs mostly in USDAA, which has always allowed all dogs to compete. NADAC and CPE are other venues that (as far as I know) have always allowed mixed breeds.
As of maybe four years ago, mixed breeds became an option in AKC agility as well. Last I knew, clubs were not required to accept entries from mixed breeds, but were allowed to. The registration will say something like "Canine Partners accepted" or something along those lines. I know here in the northeast, most, if not all, AKC trials allow mixed breeds.