[Nathan Zachary]

That is a MUST! Always get their rental history, and DO follow up on it. You'd be surprised at how may applicants lie about it, or not even bother to include some of the places they rented. Sometimes by talking with their previous landlord, You can get that information. But be careful, some landlords will say anything to get rid of a tenant. Take a drive and check out the place they are living at if possible.
Do a credit check, and call their employer. The time you take to screen a tenant is well worth it. It can save you a lot of grief.

[elmer fudd]

"Sometimes by talking with their previous landlord, You can get that information."

Yep, the previous landlord is the important one. Their current landlord might have a vested interest in seeing them move, but if they no longer live there then the landlord can give it to you straight.