Jynx is perhaps the best, largely because hair is extremely important to the jynx culture. They understand more than any other pokémon the intricacies of using hair as a form of expression, and thus, they’re far more open to listening to clients and working with them to style their hair (or fur, in the case of animals and some pokémon) in just the right way that best suits who those clients are. Moreover, even in the wild, jynx are adept at caring for hair, and as such, they’re very keen on learning and using the proper techniques for managing a client’s hair.
By extension (no pun intended), smoochum are the second best, as they’re often apprentices of more experienced jynx.
Besides jynx, shockingly, mr. mime (note: not mime jr., which is often more interested in play than work), audino, chansey/blissey (despite a lack of hands), and aromatisse are the next best, due to their eagerness to serve people in more domestic tasks—or, in aromatisse’s case, an interest in trends.
As a note, contrary to popular belief, pokémon such as scyther, scizor, and others known for sharp claws are not good choices and should never be used for such purposes. In some cases (such as sneasel), it’s because they’re fickle and may cut clients’ hair in wild, erratic patterns for fun. In other cases (such as sandslash), their type or body plan prevents them from being effective for anything besides cutting. And in others (such as scyther), it’s a combination of the two.
The Pokédex According to Bill 