Nowadays it seems like rocket-powered and jet-powered motorcycles are commonplace, with the principal domain of rockets being, naturally, sprints, where such machinery holds the current world record for two-wheel speed.
Arvil Porter built several Rocket Bike dragsters around 1975, which reached speeds of 200mph in the 1/4 mile. Some interesting problems arose, not so much during construction of these specials (after all, they're basically just a rocket engine with two wheels...), but during their use on the racetrack. First, it appeared after every run that the rear wheels had been locked the entire time... which turned out to be the inverse problem of most sprinters; rather than wheelspin, they had wheel DRAG, as the wheels weren't keeping up with acceleration! When the rocket ignition button was pushed, the tires would squeal as if they were spinning under power, but they were spinning to catch up (there is no drive through the wheels on a rocket bike).
Second issue, and much more dangerous, was the g-force affecting rider consciousness during a run; ie, the riders would often pass out from their blood flowing into their legs and away from their brain...some early remedies included using duct tape very tightly on their legs! The ultimate solution was the purchase of surplus Soviet jet-fighter pilot g-suits, which pressurize around the legs to keep the blood 'up top'.
A couple of interesting websites, if you have an interest in how Archibald Low and Fritz von Opel's ideas have developed:
Great website from the designer of Evel Knievel's X-1 Skycycle, with lots of archival photos of construction and design.
Tecnologia Aerospacial Mexicana makes rocket belts (which you can buy!), and has built a rocket sprinter, among other fascinating projects.
Rocket Powered Vehicles is a website of Ky Michaelsong, who has built rocket bikes and cars since the 70's, including one built for 'Super Joe' Einhorn (shown above), and the 'Human Fly', shown in flight below, jumping over 27 buses in 1977.