Conformation is the only thing I can think of where you'd be required to have a purebred dog. Both the AKC and UKC offer an alternative registration for mixed breeds and dogs that look pure without papers. I do think they require them to be fixed though, not 100% on that.
You can enter pretty much any of the competitions from there.
NADAC is a agility registry that only requires the handler/dog to be a member. Or so it was the last time I trialed through them. (once)
Dakota - GSD/Lab Cross - CGC CA SRD PD Mondioring OB1 Diabetic Alert Service Dog (Handler/Owner trained)
Alice- APBT mix
Ryuk- Ghost Corn Snake