"Cut" can be learned by over 250 different Pokemon in the series, with the majority of ground, poison, bug and normal Pokemon capable of learning the move. As of 2014, "Cut" is the first Hidden Machine (HM 01) in all current generations of Pokemon.
"Cut" can be learned by a huge variety of Pokemon. The best way to know if a Pokemon can learn the move is to attempt to use HM 01 on the creature, or look the information up in an online database. Though many Pokemon can learn "Cut," there are some Pokemon that can only learn it in certain versions of the game.