It has been surprisingly few; we've been dealing with "inappropriate" sigs and other issues as much as particular threads.
Normally, I prefer to close a thread rather than edit out individual posts. For one thing, it's a lot less labor-intensive for the moderator, and for another the closed thread then serves as a kind of example or warning to others -- an example of "what not to do" so to speak. The desire to not have threads closed, then, leads to members self-policing their own material and creating a bit of "peer pressure" that is (in my opinion) more effective than a squad of moderators hanging around and checking each and every post.
However, as some have noted, there have been some minor signs that some folks are purposely polluting threads with inappropriate comments, apparently for the sole purpose of getting the thread closed. So for now (and probably not for very long, really, but for the short-term) we're going to edit out inappropriate comments and e-mail the user individually as we do so (unless the entire thread is clearly out of control or on a downward spiral). Yes, this is a LOT more work for the moderators, but we'll see how it goes.