Then you are not open to all pets yet not closing off the possibility altogether....So, for example, one small well behaved dog you may approve  (especially one that a previous landlord says was not a noise, damage or other problem) but a pack of loud unruly ones not so much...And you can refine some of what you are looking for in terms of size, behavior, numbers, breeds, ages (no puppies without house training for me, eg), even spaded/neutered (especially for male cats, etc), references (ie how the owner did with a pet in past places), etc... 

When people have asked, I've generally said no very large breeds and no violent breeds. 

Type, breed, size and age matter.

.- No longer able to limit "vicious breeds" i.e.

It's why you see insurance companies so particular on what breeds they allow etc.

My agent (with Allstate) says their policies exclude any dog with a known bite history, but no specific breed restrictions.The agent strongly encouraged me to require tenants to carry their own renter’s policy that includes a minimum $200,000 liability.

In areas where it's hard to find a landlord to accept pets, or large breed dogs (not necessarily pit bulls), like the SF Bay Area, you can get that higher rent just because you're willing to take labs or greyhounds, for instance.So, you can get a higher rent.  

@Randy Lehey we filter based on breed and weight of the dog.

Moral of the story: judge the dog, not the breed.

., $35/month for 1 pet, $70/month for 2 pets).3.No aggressive breeds.