A hangnail was a loose flap of dead skin on the side of an individual's finger. They were viewed as a minor infliction, not worth the time or trouble of medical attention from a physician.
While stationed aboard USS Discovery in the late 2250s, Jett Reno reported to sickbay with a hangnail which she claimed hurt "like a bitch", as a pretext to talk to doctor Hugh Culber. (DIS: "Through the Valley of Shadows")
In 2267, when informed that he was needed aboard the USS Enterprise due to an unspecified "medical emergency", Doctor McCoy dismissed it as somebody having discovered a hangnail. Regardless, he beamed back aboard the Enterprise, promising to tell Kirk later what it was that had been so urgent. (TOS: "The Menagerie, Part I")
In 2370, while receiving an extensive annual physical from Dr. Julian Bashir, a replicant of Miles O'Brien told him that if he couldn't leave until the doctor found something wrong with him, he'd try to grow a hangnail to satisfy Bashir. (DS9: "Whispers")
When Kes asked The Doctor for nitrogenated soil samples for the hydroponics bay in 2371, he waxed melodramatic, as this was the latest in a long line of (in his opinion) trivial requests USS Voyager's crew had made of him, complaining, "Now I know how Hippocrates felt when the king needed him to trim a hangnail." (VOY: "Parallax")