Good question. I was just talking to my insurance agent about it this morning.
My insurance company won't cover the house if it is certain dog breeds(surprisingly dalmatians are on that list) or if it has bitten a human before.
I always have a $300 pet fee - non-refundable- that goes along with the security deposit. Additional rent is also typical. For where some reason, people love their pets and often have trouble finding other places that accept them. Unless you have an extremely hot market where you are extremely selective, why cut out 50% of the applicants?
I don't allow cats. If you do, take a black light around your baseboards when they tenant moves out, as well as check the HVAC vents. Pretty disgusting. Obviously, the same issue could happen with dogs, I've just seen it more prevalent with cats. After the second time, I stopped allowing cats. For some reason, dog owners tend to be a little more cognizant of the issue.