I currently accept any pets (up to 4) without a fee in my listing, but I am considering adding language prohibiting cats, because I have lost a few potential bookings to people who have cat allergies. (Which I understand, even though I am obsessive about cleaning after each guest, and store all cat supplies in the basement when not in use.)
It's really difficult to find a good spot in our small guesthouse to place a litterbox. The first "cat guest" I hosted shredded the fabric cover on the bottom of one of the bed's box springs. After that, I added a scratching post for when I know a cat will be visiting and that helped. I can tell when cats (or dogs) have been on the beds, which is prohibited (we provide pet-friendly bed covers that guests can use if they are unable to keep their pets off the beds so that I don't have to wash the bedspreads quite so often).
The main thing that has kept me from prohibiting cats is that I don't want to add that restriction to my listing title (seems negative), and I fear that guests will ignore it anyway (as they do with much of the information I include on the Airbnb site).