Lots of great advice here, so I will share my story. What I've learned is that it is not about the tenant, but the property COMBINED with the tenant.
If the property will rent to better paper tenants then spend little time vetting them, if not then keep reading.
We have a property in a not so hot rental area (old house under water) and were getting plenty of people interested, but most were either weak to begin with or would have blemishes on their background. As we all know, if they were as brilliant and resourceful as we are then they would all be owning a duplex and living in one unit and renting the other. That obviously is not the case with many, many people. Credit scores can get wrecked for many reasons, but too many people have no idea of what the impact is.
The tenants we eventually rented to are a couple that lost their house and were in a struggle. They even had difficulty showing any kind of income, let along the amount they needed for the house. In the end, it came down to my gut, but my gut based on many tests (and desperation to find a tenant - took 6 weeks), which included asking as many questions that I could think of and requests for many documents and information. Each was a test and gave me more and more confidence that they were telling the truth and really wanted to be there. How quick they responded, what they provided and the clarity of their story was huge. The end result is that they are still the tenants that take care of the house better than I did or any of my other 50 tenants!
Bottom line is that you have to get to a level of comfort that they can afford the place and are responsible enough (as individuals) to warrant an approval. This level of effort to place tenants drops with the more units you have though, as time and time to "being nice" is reduced dramatically.
On the flip side we have tenants that we followed our processes without regard to the person, but now have some good paper tenants that we can't wait to leave.
Sergio