If it can be done on a mule, it seems like it was happening at the Montana Mule Days in Drummond. My wife, Maxine, and I made it down to watch last weekend, and definitely got our gas money back in entertainment value (even if it rained most of the time). If you have a mule days event within a couple of days drive, I highly recommend you take the time to go. Here is a partial list of some of the country's Mule Days Celebrations; let me know if you know of any more that I should add.
Competitive classes ranged from formal halter and Western/English equitation classes to driving classes to reining, roping and cattle events. There were events for the kids to compete in, such as breakaway roping, barrel crawl through, the balloon bust and "Lead your donkey/mule to water". Of course, it was Mule Days, so there were a lot of fun events you won't find at a traditional horse show or o-mak-see. Traditional classes are the log pull and the 4-up hitches. My favorites (because I wasn't competing in them) were the Flapjack race and the Packer's Scramble.
The Flapjack Race is really what it sounds like. The show supplies wood and matches. The contestants supply everything necessary to make a flapjack - the batter may be made ahead of time. No paper, charcoal lighter, starter fluid or flammable materials are allowed. All contestants line up beside their animals and at a whistle, lead a pack animal to the opposite fence. At the fence, they each unpack the animal, build a fire and cook an edible flapjack at least 4" in diameter, all the while keeping your UNTIED animal within 15 feet of the fire. The unfortunate judges have to determine if the pancake is edible.
The Packer's Scramble is also a timed event, with a pretty hefty cash pot and trophy buckles. Each team consists of 2 packers and 4 head of stock, of which at least two must be mules and must be packed. The contestants must supply the following: axe, bucket, shovel, 4 or 5 manty tarps, ropes, and normal saddle gear. The teams enter the arena, unpack their loads, turn all animals loose, and crawl into their "sleeping bags" (manties). At the signal, the contestants have to get up, catch and saddle their riding horses and pack the mules with all of the gear - plus a live chicken. They run an obstacle course with one packer leading the 2 mules pigtailed together. The other rider must be mounted as an outrider. The fastest time wins - if you arrive with all of your packs intact. I did manage to take a few pictures of this event - enjoy!