Most Pokemon can learn Cut. Some easy 4th Gens are Bidoof or Bibarel, Budew, and Pachirisu. A full list can be obtained at bulbapedia.org
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your Pokemon CANT learn cut in ruby, sapphire or emerald. you have to trade a Pokemon that's holding the TM cut (from sinnoh region game, platinum, diamond and pearl) then take it off them and they can learn cut.
only Pokemon with sharp claws can learn the hm cut
No, it is not able to learn cut.
Grass Pokemon
most Pokemon can learn this, to name a few there are rattata, ratticate, sudowoodo etc and loads more.