A motorcycle was a two-wheeled engine powered land vehicle, developed on Earth in the late 19th century.
Motorcycles were used during World War II by the military of Nazi Germany. Civilians also used motorcycles in the 1980s and 90s.(ENT episode: "Storm Front"; TOS movie, novelization & comic adaptation: Star Trek IV: The Voyage Home; VOY episode: "Future's End")
A PX70 motorcycle was kept aboard the USS Franklin by a member of the crew. Starfleet Lieutenant Commander George Kirk owned a PX70 motorcyle as well. According to his son, he would take his wife Winona on it and drove her crazy.(TOS movie: Beyond)
Starfleet Ensign Thomas Odell owned several restored gasoline-powered motorcycles, which were a favorite conversational topic of his. (ST video game: Elite Force)
While recovering from his infection by an Ardanan parasite, Fabian Stevens had a dream about being chased by men in white robes wanting to turn him into a motorcycle. (CoE eBook: Signs from Heaven)
In the Nero-created alternate reality, James T. Kirk owned a motorcycle as a young man in Riverside, Iowa, which he abandoned when he joined Starfleet Academy in 2255. (TOS movie: Star Trek (2009))
In 2258, a mutated Gary Mitchell referred to the vehicle as a motorbike. (TOS comic: "IDW Star Trek, Issue 2").
In 2263, Captain James T. Kirk found a motorcyle in the USS Franklin on Altamid. Kirk and Jaylah then used it to distract Krall and his drones while McCoy and Spock freed what was left of the USS Enterprise's crew.(TOS movie: Beyond)