As Realtors representating several landlords, we are first interested in knowing the creditworthiness of the prospective tenant. Once we ascertain their ability to pay the rent, then we can consider the creditworthiness of their pets. We would want to know the breed, the age, the size, a letter from the vet confirming recent shots and then, after some preliminary research, decide if it is worthwhile for the landlord to accept a particular pet.
Once a pet is accepted, the lease will address issues that may arise by allowing a pet.
Also, it may be advantageous to have an insurance policy to cover any problems that a pet may cause. If a dog bites a guest visiting a rental home then there are liabilities.
Pets can be great tenants, but we must do our 'due diligence' to make sure they stay that way both landlord and tenant!