Is she vaccinated?

Here's some quick tips:Credit history - no collections, repos, late paysIncome - 2 1/2 times to 3 times the rent, verified by pay stubs and a call to their human resources department at their workRental History - Calls to the prior landlords and ask "Was the tenant given any three day notices during his tenancy with you and if so, what were the dates and what was the reason.For assistive animals/pets, require them to be vaccinated, spayed/neutered, get a letter from a third party who is a social worker, medical doctor or psychiatrist to explain the need for the animal by the tenant.  

Then you make sure that you like the individual pet, that it is well mannered, spayed/neutered, clean, flea-treated, up to date on its vaccines, and has good references from previous landlords.  

.- Require all animals be registered and vaccinated.

I do specify what kinds of pets are accepted (dogs and cats only, no known bite/attack/aggression history, proof of vaccination, etc.).

 - She is offering to pay more pet fee (I am already charging $200 upfront and $25 additional every month)- The house is old (1952), has hardwood floors, a new deck, and a large backyard- The tenant is required to get renter's insurance- I have asked her to provide a record of the dog being spayed/neutered as well as vaccine recordsIt may or may not be obvious that I would like to approve this tenant even with the dog.  

Asking to be provided proof of recent vaccinations of the pet will also help ensure you have a responsible pet owner.And as @Steve Vaughan stated above, small dogs tend to do the most physical damage.

They gave me a pet deposit, but I subsequently found out when a possum came in the house that the dog wasn't vaccinated for rabies.

Unfortunately, many of these doctors offer their services on line to provide this certificate with letter for a fee.You can accept the animal under reasonable conditions, such as require that the animal be registered and that the tenant show proof of current vaccination and that the animal is not an aggressive threat to others. 

If you prefer to let your cat out, it has to be supervised and leashed.Dogs : We allow adult well trained and well cared for dogs, they should be vaccinated up to date, clean “bite record from the vet” is required.