Can I use cheap vodka in cocktails?

For cocktails that mix vodka with strong-flavored ingredients (Moscow Mule, Bloody Mary, Espresso Martini), yes. For Martinis, where vodka is the lead, use a mid-tier bottle at minimum.

Vodka is, by legal definition, supposed to be flavorless — distilled and filtered until almost neutral. In practice, cheap vodkas have rough edges (acetone notes, alcohol burn) that disappear behind strong mixers and stand out behind weak ones. Decision rule: if the recipe has a dominant flavor (ginger beer, fresh espresso, tomato juice, large amounts of citrus), $15 vodka is indistinguishable from $40 vodka in the glass. If the vodka is the dominant flavor (Vodka Martini, Vesper, vodka soda), you can taste the difference clearly. Mid-tier bottles for $20 to $30 that work for any application: Tito's Handmade, Ketel One, Stoli, Reyka, Wyborowa. Higher-end if vodka is your spirit of choice and you'll feature it neat or in Martinis: Belvedere, Grey Goose, Chopin (potato-based, distinct). Skip the under-$15 plastic-bottle vodkas for any cocktail you would actually serve a guest — even the strong-mixer drinks taste like they're trying to hide something.