We use MySmartMove.com
We do not charge an application fee.
This is our process:
1) Give their application the sniff test for income, work history, pets, and other questions you might care about.
2) If they pass that then we call and verify their employment, previous landlords, and references
3) After all this if we still like them then we tell them they are approved pending a background check. We collect their email and send them a background check that they need to complete through MySmartMove. It costs them $35 and we pass that cost on to them. If they pay it, we know they are committed.
4) We either approve or reject.
There may be better screening processes or services but we have found success with this.
Hope that helps.