In AmE, a punctured tire is referred to as "flat." A person might return to his car at the end of the work day and find one of his tires flat. When he changes the tire he'll notice that there is a nail stuck in it. Technically, the nail has punctured the tire, but we rarely describe a flat tire as a "puncture" in AmE.
And, as Dave mentioned, if a vandal purposely flattens your tires, we say that they have been "slashed."