We have the benefit of our rental being our basement. So we would know if someone tried to sneak a pet in long-term, though we did have someone ask if they could dog-sit for a friend for a week once.

We say no pets up front. Because we live here, we tend to get all of our tenants through word of mouth - kids of friends or acquaintances who are getting married, etc. So there generally aren't any surprises, which is nice.