A handkerchief, hanky or tissue was a small square of fabric or paper that could be used for personal hygiene, such as blowing the nose or wiping one's hands. It was generally carried in a pocket and may have been plain or patterned.
When an area of Cold Station 12 began to suddenly be gassed in 2154, a startled Doctor Jeremy Lucas hurriedly took a handkerchief out of a pocket and used it to cover his mouth and nose. This use of a handkerchief gave Lucas just enough time to activate a code black security alert inside the station, though he eventually collapsed. (ENT: "Cold Station 12")
Waving a handkerchief was a symbol for an emotional goodbye. In 2257, Queen Me Hani Ika Hali Ka Po indicated to Sylvia Tilly that she intended to stay with Tilly and the USS Discovery to defend Xahea during the Battle near Xahea with Control's forces, noting: "What am I supposed to do, wave a hanky from my throne?" (DIS: "Such Sweet Sorrow")
When Commander Benjamin Sisko left Deep Space 9 to visit Bajor, he informed Li Nalas that Lieutenant Dax and Chief O'Brien could handle Ops and if he needed any further help, but Li made a joke and said he could not sneeze without three people handing him a handkerchief. (DS9: "The Circle" Keiko O'Brien once gave Miles a silk handkerchief scented with her perfume. (DS9: "Rivals")
In 2373, Leeta dismissed a Risian traditional wedding dress she was shown as consisting of "two handkerchiefs and a loincloth." When her fiancé Rom suggested not wearing the handkerchief portions of the dress, she refused to be naked at her wedding. Rom reminded her that she'd still have the loincloth. (DS9: "Call to Arms")