In my years in the ministry, I've seen many different kinds of congregations (after all, I've spent more than 12 years as a circuit counselor), but the kind that presents the most problems may not be the ones you think. Congregations in conflict would seem to rank high on the list of problems, and they do present their own unique difficulties. Often, though, a little bit of patient listening and some fairness and impartiality can take care of many of the conflicts.
To my mind, the biggest problem is presented by the self-satisfied congregation – by the congregation that is happy with things just the way they are, and has no desire (and usually no willingness) to change. Often such congregations are perceived as being unfriendly and unwelcoming, whether they really are or not. I remember talking to one of the leaders of one such parish and suggesting they improve their parking area (because they had very little off street parking). The response I got from this leader was basically that new people should park blocks away and walk the rest of the way to church – the reason for this? Well, the members of the congregation have done it that way for 80 years, so new people should do it too.
It's unfortunate when congregations get to that point. Often there seems to be very little a new pastor can do to change such a congregation. One district president I've known over the years told me that that the only thing you could do with a congregation like that would be to leave it vacant and hope that people would finally come to their senses. As I said, it's sad when a group of Christians reach that point.