As to the question of "who owns the house?", which one of you wants to own the house? If the answer is neither, sell it, and move on.

Refinancing the home, pulling the equity out to pay off the other, may be a lot more difficult at the moment because of changes in mortgage regulations, etc. Especially true if there are credit issues.

If credit is clean, and there is sufficient lendable equity to take care of the other person, there shouldn't be an issue. Also take into account the rate now, and what the rate will be after refinancing.

If credit is still crappy, and/or there is insufficient equity, it may not be such a simple solution. Sale of the house may be a better option.