Preparation of the martini generally consisted of a small amount of vermouth poured over ice to slightly melt and flavor the ice. The desired amount of gin or vodka was added, and the mixture was shaken or stirred according to taste and then served without the ice.
Philip Boyce, after being summoned to Captain Christopher Pike's quarters, noted that "no one wants a warm martini". When asked why he thought Pike needed one, Boyce commented, "Sometimes a man will tell his bartender things he'll never tell his doctor." (TOS: "The Cage", "The Menagerie, Part I")
Vic's Las Vegas Lounge served martinis. When Jake Sisko and Kesha joined Nog at Vic's, Kesha ordered a martini. Julian Bashir ordered a vodka martini, stirred not shaken. (DS9: "It's Only a Paper Moon", "Badda-Bing, Badda-Bang")