(covers information from several alternate timelines)
Wheel on an abandoned bicycle on Terra Nova
A wheel was a round device capable of rotating on its axis, facilitating movement or transportation, or performing labor in machines.
A spinning wheel was a wheel that was used to produce yarn. (TNG: "Up The Long Ladder")
Wheels were essential to the drive of automobiles. As they were a key part of the movement of cars, trucks, vans and other automobiles. There were typically four wheels on the bottom of a vehicle, used for driving it, and a steering wheel used for changing direction. A tire lock was attached to wheels to prevent the theft of automobiles. (ENT: "Carpenter Street")
Tire irons were used to remove wheels from an automobile, which was necessary when changing the tire. (Star Trek IV: The Voyage Home)
Other vehicles or devices equipped with wheels included wheelchairs, wagons, bicycles, motorcycles, tricycles, tea-carts, and gurneys. (ENT: "Terra Nova")
A bird on wheels
In 2151, a statue depicting a bird on wheels was found in the atrium of the Vulcan monastery at P'Jem. (ENT: "The Andorian Incident")
In some Native American mythology, a medicine wheel was used to guide a person's spirit. (VOY: "Cathexis")
There were also games that made use of wheels, such as roulette. In the game, a wheel is spun on the game table, leaving it to chance as to where the wheel stops and the ball lands. (TNG: "The Royale")
A common saying in dabo, a Ferengi game similar to roulette, was "watch the wheel, not the girl." (DS9: "The Abandoned")
Tongo was another Ferengi game that made use of a "Tongo wheel." (DS9: "Business as Usual")
When Trip was trying to describe a car to Kaitaama, he told her it was a vehicle with four wheels, and that it wasn't much bigger than an escape pod. (ENT: "Precious Cargo")
In an alternate 2259, Admiral Christopher Pike attempted to explain to Captain James T. Kirk that the Nibirans were especially vulnerable to cultural contamination, as they had "barely invented the wheel." (Star Trek Into Darkness)
That same year in the prime reality, La'an Noonien-Singh sang a song that had the lyrics "If only I can let go of the wheel." (SNW: "Subspace Rhapsody")
In the 2260s, a woman named Mavig was a musician that played a harp that was shaped like a wheel. (TOS: "The Way to Eden")
When Hikaru Sulu was telling the rest of the senior staff of the USS Enterprise about the USS Excelsior having transwarp drive, Montgomery Scott told him "if my grandma had wheels, she'd be a wagon." (Star Trek III: The Search for Spock)
The song "Just in Time", sung by Vic Fontaine, had a lyric referencing the wheel: "Just let the wheel of love go round." (DS9: "It's Only a Paper Moon")
In the 2370s, the Argo was a shuttlecraft attached to the USS Enterprise-E that was equipped with a buggy that was also called the Argo. It was held in place in the shuttle by clamps attached to the wheels. (Star Trek Nemesis)