A motorcycle was a two-wheeled engine powered land vehicle, developed on Earth in the late 19th century.
A motorcycle was kept aboard the USS Franklin by a member of the crew. Starfleet Lieutenant Commander George Kirk owned a motorcyle. 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 to distract Krall's drones while McCoy and Spock freed what was left of the USS Enterprise's crew.(TOS movie: Beyond)