To make the peppermint vodka: add vodka and candy canes to a jar. Seal and let sit until candy canes are dissolved (about 2-4 hours). May make up to 1 week ahead; store in refrigerator.
Rim 2 martini glasses with crushed candy canes: dip the glass in water then in a pile of crushed candy canes.
To make the Candy Cane Martini: Add both vodkas, schnapps, and creme de cocoa in a cocktail shaker with ice. Shake and strain into martini glasses.
To make the Creamy Candy Cane Martini: add vodka and Irish cream to a cocktail shaker with ice. Shake and strain into martini glasses.