Actually, I don't think there's a need to store multiple hashes.

Here's one idea: Let's say the user's password is P. The user enters some password P' with a typo. The authentication check is "does H(T_k(P')) == H(P)" for some set of transformations {T_1, T_2, ..., T_n}. Each transformation T_i hypothesizes that the user made a specific mistake. (e.g., T_1 is the caps lock is on so we need to flip the case of all the characters)