(covers information from several alternate timelines)
Human hair cut in a Vulcan fashion
A beehive in progress
A haircut, hairstyle, or coiffure was a specific fashioning of an individual's hair; the difference lay in whether it was trimmed into a particular shape (cut) or merely arranged differently (style).
One could have one's hair parted on the left, the right (more on one side than the other) or in the middle. In 2152, after apparently offending the Kreetassans for the second time, Jonathan Archer suggested that maybe his hair was parted on the wrong side. (ENT: "A Night in Sickbay")
Many races favored specific types of hairstyles or cuts, such as Vulcans, who often had a bowl cut they happened to share with Human 20th century comedian Moe Howard (ENT: "Carbon Creek"), and Romulans, who had a similar cut with pointed fringes overlapping their forehead. Stefan DeSeve wore his hair in the Romulan fashion following his defection to the Romulan Empire. (TNG: "Face of the Enemy")
Some hairstyles incorporated bangs, a section of short hair combed over one's forehead. In 2259, James T. Kirk vented about his frustration with Spock to Nyota Uhura, expressing his desire to "to rip the bangs off of his (the Vulcan's) head". (Star Trek Into Darkness)
In 2267, Khan Noonien Singh found Marla McGivers' hairstyle unbecoming. When he asked her why she wore it that way, she claimed it was comfortable. Khan proceeded to undo it. (TOS: "Space Seed")
One popular style was the beehive, favored by Starfleet women in the 23rd century. Jadzia Dax wore her hair in a beehive to blend in while time travelling to 2268 to help thwart "Barry Waddle"'s plot to kill James T. Kirk. (DS9: "Trials and Tribble-ations")
When Geordi La Forge visited the barbershop aboard the USS Enterprise-D in 2367, the Bolian barber told him he didn't need a haircut as he was in last week. (TNG: "Data's Day")
When Worf visited the same barbershop two years later, Mot asked him about a new haircut or a beard trim. (TNG: "Schisms")
When Odo assumed the shape of a humanoid, he imitated Doctor Mora Pol's hairstyle. (DS9: "The Forsaken", "Necessary Evil")
When Tom Paris was a child he wore a cap to cover his haircut he got from his father on the first day of every summer. (VOY: "Faces")
In 2378, Seven of Nine asked B'Elanna Torres how she achieved her "appealing coiffure". Torres replied that it was a combination of a sonic shower and a little engine grease. (VOY: "Human Error")
While on the surface of the Ba'ku planet in 2375, Captain Jean-Luc Picard told Worf he needed a haircut. Worf experienced jak'tahla and told Picard that accelerated hair growth was normal during this time. (Star Trek: Insurrection)