I allowed pets (three suites), and they were all great.
In BC, we can charge double the security deposit when a pet is included, so I usually charged that, but almost every tenant-with-pet got that back.
I required the pet to come to the viewing/interview. That way, I could see a lot about the human (how they interact with a vulnerable, dependent creature), see the animal's behaviour (activity level, respect for furniture/flooring/etc, barking, aggression/gentleness), see/smell cleanliness.
I rented to gentle, quiet animals with gentle, quiet humans.
The only pet-related issues in those tenancies were:
1. One dog -fully indoors, even litter-trained- shedded like crazy and left a thick layer of fur everywhere when they moved out. I took some off the deposit for all the cleaning that required, but no permanent damage.
2. One tenant with a very large dog had an accident and became unable to walk her dog and pick up his poo from the yard. The yard was shared amongst all tenants, including children who played in it. I did it for a while, but eventually said that if she couldn't find someone to provide this care for her dog, either the dog or both of them had to move on.