They must be an approved breed (not on my insurance companies blacklist) and provide documentation showing breed, up to date on shots, and licensed.

One prospective new tenant has a service dog, that I believe is a dangerous breed, or a mix with a dangerous breed (pitbull). 

With older (and many not so old) buildings, it's almost impossible to prevent their entry, but it is possible to make the building less attractive of a place to stay.Mice breed quickly, so ultimately they become the property owner's problem regardless of how they entered and whether the tenant laid out a buffet enticing them to stay.  

A bunch of Home Owner Associations here in Houston, TX have restrictions on Breeds as well. 

Just make sure you don't allow puppies ; kittens ; or hyper/destructive breeds and you should be fine

I wouldn't even worry about using the type of breed excuse.

@Chris Kelly Someone sent me a list of Excluded Breeds of Dogs by their Insurance Company.

We're doing it for the first time and got the details about the restricted dog breeds from our home insurance agent. 

Likewise, you can't restrict breeds, sizes, require alteration, or other such requirements.

. - Restrictions on dog breeds, - Restrictions on the numbers of properties insured with 1 company- LLC's or entities on personal policies,  - Vacant or properties to be rehabbed- Replacement or Actual Cash Value or Agreed Value, - Bundling policies for additional discounts- Umbrella policy requirements, do they allow other carriers?