I was a ring steward at a local dog show in October, and mixed breeds competed in both obedience and rally (there was no agility trial at this particular show; the local AKC chapter does that in the spring). The only thing in which they could not compete was conformation, for obvious reasons.
In addition to all the other things already listed, mixed breeds can also certainly do therapy work as well. The AKC now awards titles in obedience, rally, agility, and therapy to mixed breeds too, if you're interested in that.
AKC Canine Partners can compete in Agility, Obedience, Rally, Tracking and Coursing Ability as well as AKC Canine Good Citizen, AKC Canine Good Citizen-Advanced, AKC Therapy Dog, AKC Search & Rescue Dog, Barn Hunt Association, North America Diving Dogs, and North American Flyball Association.
It's also enormously easy to register your dog as a canine partner with the AKC. Since you're not trying to register your dog as any particular breed, you don't even need to supply a photograph and can do the whole thing on line (though I do think the "certificate of fun" they send you for registering your dog is a bit silly and looks like something you'd "award" to a child; I don't know why the paperwork couldn't look a bit more sophisticated, since as long as the dog is identified on it as an "All American," there's no way anyone is going to be able to pass the dog off as a particular breed). The fee is relatively minimal as well, but by registering your dog as an "All American," you can then get the certificates for CGC etc. (which look more like what I wish the "certificate of fun" looked like---and check that "certificate of fun" when it arrives, as they had my male dog listed as a female, though I certainly hadn't registered him that way, which they agreed I had not and sent me a certificate; I really didn't care about the certificate, but I wanted to make sure the details on his registration records were correct). As an academic myself, I wanted the certificates for the CGC and CGCA.
I think tons of people want mutts. And then tons of people also want 'designer dogs' which is just a mutt with a fancy name.
What would you envision would happen at a mixed breed dog show? At a normal conformation show, dogs are judged for how well they meet their specific breed's standard. Mixed breeds can't have that, they don't have a standard. I suppose you could create some generic awards such as 'biggest dog' and 'smallest dog' but most people's dogs would be out of the running. Maybe cutest dog, but that is way too subjective.
My community has a yearly 'pet parade' where people walk their dogs and other animals down a given path and people come out to see all the animals. I think an event like that is more useful than a 'dog show', no awards are given, but you do have a fun time with other animal lovers.
Also, there are plenty of competitions that mixed breeds can participate in, and even win. Obedience, agility, search and rescue, herding, frisbee catching, heck even dog-human dance contests. If you have a dog that is amazing at all these random things, seems to me you'd have a dog that would excel at one of those contests. Being a mixed breed isn't going to hold him back
It's already been said, but if you want the dog registered (and you're American) you can register the dog as a Canine Partner. My mixed breed Stella is registered with the AKC as an All American and has her Puppy Star medal, and we are working towards Canine Good Citizen. Also, every single dog sport and activity-with the exception of conformation (fancy dog shows) are open to mixed breeds, even AKC events. And trust me, the others are much more fun, with much happier, healthier dogs and much nicer people. No offense to anyone who might do conformation on here but most show dog people are um, a little crazy, snobby, and not so friendly. Basically, that movie "Best in Show" might as well be a documentary!