19th century dental signage
22nd century dental instruments
Dentistry is the study and professional care of a person's teeth. A dentist is a person who practices dentistry.
Most Starfleet medical personnel are trained in numerous fields, including dentistry. But there is also a dedicated sub-division of Starfleet Medical, Starfleet Dental, responsible for Dentistry. (DS9: "Paradise Lost")
Dentists
In the 19th century on Earth, Doc Holliday was the local doctor and dentist of Tombstone, Arizona. (TOS: "Spectre of the Gun")
In the 20th century, Maury Ginsberg was a very successful orthodontist in Scarsdale, New York. Orthodontics is the application of "braces" on a person's teeth, to realign teeth that would naturally fall out of alignment. (VOY: "Death Wish")
In the latter-half of the 20th century, time-traveler Dr. Leonard McCoy was sarcastically accused of holding a degree in dentistry by a surgeon preparing to operate on fellow temporally-displaced patient Pavel Chekov at Mercy Hospital. (Star Trek IV: The Voyage Home)
Dental practices
Doctor Phlox had many degrees in alien medicine, among them was a degree in dentistry. (ENT: "Dear Doctor", "A Night in Sickbay")
On Vulcan, a common dental practice is for a Vulcan to have their teeth coated in a trifluorinate compound to prevent cavities. Despite having this done in 2128, T'Pol had a cavity in one of her tricuspids that Dr. Phlox had to restore in 2151. (ENT: "Dear Doctor")
The inhabitants of planet 892-IV utilized a dental hygiene product known as Mars Toothpaste in 2268. (TOS: "Bread and Circuses")
In the Ferengi and Klingon cultures, teeth are sharpened with a tooth sharpener. (DS9: "Little Green Men")
Background
In the novel, Chain of Attack, a Dr. Rajanih was the doctor in charge of dentistry on board the USS Enterprise.
Concept art for the USS Enterprise-D had a dental ward adjacent to sickbay.