Aquiel Uhnari wearing a wig
A wig, hairpiece, or toupée was an artificial arrangement of hair worn for fashion, disguise, or cultural reasons.
Vulcan Sub-commander T'Pol wore a long-haired wig during an undercover mission to the Akaali homeworld in July 2151 to hide her Vulcan ears from the pre-warp civilization of the Akaali. (ENT: "Civilization")
Three musicians of an hallucinated string quartet, seen by a crewmember, were wearing period wigs. (TNG: "Where No One Has Gone Before")
T'Pol wearing a long-haired wig
At one time, it was fashionable for Betazoid women to wear elaborate wigs that caged small animals. The practice, which was cruel to the animals, was stopped when one "formidable woman" stood up against it. (TNG: "Half a Life")
This was a real fashion in Human history, too. It was also a plot device in the Lucille Ball/Bob Hope movie Fancy Pants.
Lwaxana Troi used to wear wigs from time to time. (DS9: "The Forsaken")
In 2368, Mister Mot fitted Data and Jean-Luc Picard with hairpieces as part of their Romulan disguises. (TNG: "Unification I")
Aquiel Uhnari's sister Shianna always thought her sister would look good with a wig. Uhnari tried it out in 2369 but was not too impressed, as was Geordi La Forge. (TNG: "Aquiel")
While trying to come up with character sketches for possible stories, Jake Sisko once watched a Bolian from the upper level of Deep Space 9's Promenade, and speculated that he was wearing a toupée in order to impress a dabo girl. (DS9: "The Muse")
Dala wore a wig while impersonating Kathryn Janeway. (VOY: "Live Fast and Prosper")
In the final draft script of TOS: "Miri", Jahn was described as wearing a somewhat-askew Beatles-esque wig, with "the price tag still dangling."
Actor Walter Koenig wore a wig during his early appearances as Pavel Chekov to emulate the hairstyle of Davy Jones from the Monkees.