In fact, cheap enough, be sure they're checking criminal, evictions, employment, landlords, credit, etc.

Criminal and civil charges will not help that.

These requirements might not deter a criminal or even an irresponsible person from having a gun, but it could give you additional leverage to terminate a lease or limit your liability if a problem occurred and you could show that the tenant was violating the lease.

Here is an excerpt from our rental criteria.LEGAL/CRIMINAL HISTORY1.          

I can see criminal cases and civil cases.

Don't be cheap on screening - do background, criminal, credit, employment, eviction.  

The results of the criminal trial are irrelevant to you as long as they are found to be at fault.

I used to put in my ads "All applicants will undergo credit/eviction/employment/background/landlord/criminal checks. 

And she has no criminal history.

If they give you a phone number for their work or supervisor, google the business and ask for the HR dept, instead of using the name and number they gave you.Any lies or discrepancies they give you are valid reasons to deny them.Pay for every single type of check you can do on them, credit/background/criminal/eviction/identity, etc.