I have rented before and used screening service called TVS, Tenant Verification Service to pull credit, rental and criminal reports.I have not pulled a report in a while and they have apparently changed the way they operate, in that the renter now pulls their own reports and supplies them to me.

I doubt you'd want to send someone over with a criminal background or that is a registered sex offender.

Make sure their criminal history checks out.Another key factor is employment history.

I get a great picture of my tenant's credit and criminal profiles.  

One streeteasy.com writer made an interesting point: "This is not the criminal justice system where there are multiple degrees of culpability.

Such as criminal history or bad eviction history.

Also, always do proper screening complete with credit, criminal and eviction reports on your potential renters.  

Last time, I checked with my screening service, which gives me a score of 1 to 100 for the applicant, taking into account credit as well as criminal and eviction records.

Yes, it provides criminal background and credit check, but it doesn't ask the tenant to fill out other basic info that would be requested on a rental application - like employment history, pets, previous landlord references etc., even if they aren't checking on those items, I want that info. 

. §§ 90.396, 90.40324 hoursViolence or threats of violence by tenant or a guest; intentionally causing substantial property damage; giving false information on an application within the past year regarding a criminal conviction (landlord must terminate within 30 days of discovering the falsity); committing any act “outrageous in the extreme” (see statute); intentionally or recklessly injuring someone (or placing them in fear of imminent danger) because of the tenant’s perception of the person’s race, color, religion, national origin, or sexual orientation; second failure to remove a pet that has caused substantial damage