Spiderman could once he got to know about them. Wolverine would be able to take down the more physical villains, but the Joker (an intellectual) and Clayface (amorphous) would be too much for Wolvie to handle (since he is the slicing equivalent of the Hulk at times).
Peter Parker on the other hand is smart. He'd be smart enough to know that you do not attack Clayface directly for example, and he would not underestimate a smiling clown.
Wolvie would.