Preparation of the martini generally consists of a small amount of vermouth poured over ice to slightly melt and flavor the ice. The desired amount of gin or vodka is added and the mixture is shaken or stirred according to taste and then served without the ice.
Vic's Las Vegas Lounge serves martinis. When Jake Sisko and Kesha joined Nog at Vic's, she ordered a martini. Julian Bashir ordered a vodka martini, stirred not shaken. (DS9: "It's Only a Paper Moon", "Badda-Bing, Badda-Bang")