Pre-screening includes: income requirements in very plain terms (annual income) as some ppl don't seem to comprehend what 3x rent really means, no smokers, pets, etc.  

No evictions, No felonies, No pets, He makes 4.5x rent.Career Disabled Veteran has a nasty divorce which leads to Chapter 13.Tenant re-marries 5 years later and spouse has bad credit and is not working.Prospective tenant gets veteran retirement and disability of which 1/2 is garnished for ex-spouse.Prospective tenant has now taken a job teaching ROTC at the local high school.

For instance, I don’t allow pets in my rentals. 

You can also have a rule that says you do not rent to people with pets or that you do not rent to lawyers.Be consistent with the rules.

Maybe you can add $50/mo for a pet or per pet?

Then they got a cat lady in afterwards (no pet policy in my rental).

If you don't allow pets in your rental maybe you could add something around the yard that would repel his dogs?

My personal residence is on the edge of Phoenix and the natural predators definitely keep any cat problem from starting (don't let your pet outside if you value its life).There are several residents that leave food, water and shelter out for the cats; which doesn't help. 

Unfortunately the location isnt the most sought after but I cured that by catering to the people who care about the environment and by making it a pet friendly environment with a 3 acre dog park.

I think i am going to start requiring it for tenants with pets, if for nothing else for liability coverage in case their dog bites someone...My insurer offers discounts if renter carry insurance.