@Robert Ablorh If the collections were medical and closed, and their current credit, criminal, and eviction histories as well as their rental references are good they sound like a decent pick to me.  

On background/credit search, they both have good credit history but they both also have criminal background.

It's fairly common to require a co-signer for college students though since their income isn't going to meet your criteria and you then run their parents credit and use that as your basis for qualification.For prior evictions and criminal, those are still extremely important to run on the students.It's also quite common for there to be multiple students sharing a house, so handling split-responsibility for the payments is also another very common thing to do in college housing.

You also need to know if either has been evicted or has a criminal record.

I explain the process for moving forward - credit/criminal/income verification etc...within about a half hour, they are calling asking about where to drop off applications.  

Then, maybe you can get them out for criminal behavior?

., is he a tenant with rights you need to evict or a criminal trespasser that can be removed by police?)

You always need to run a background check on the individual applying as well, especially a criminal background check.

Figure out your criteria (accept pets, how many people, criminal OK, etc), how you are going to process applications (on-line service, fax in their application form, self serve), etc.As far as renting another place, it shouldn't hurt you - you have proven yourself financially responsible, that you are able to take care of minor maintenance issues, and probably have more realistic expectations.  

Screen for everything - credit, criminal, evictions, check all references.