Many criminals avoid places if they know there is a chance of getting caught.

It's hard to describe but after dealing with so many criminals you kinda get a feel for people.

Since you didn't mention it, I'd also recommend you do a criminal background check with a signed (real or digitally) application that gives you permission to do so.

Criminal histories-- Prior evictions-- Criminal convictionsCredit / Payment history-- Is your credit score at least 740+-- Have you filed for bankruptcy recently?

Also, your estranged wife did not legally "trespass" as she is co-owner, and even without giving notice it's not a criminal offense for her to be there.I get it, have been divorced (more then once) myself plus decades of call like this on ABC (you pick subject) and it all boils down to the same thing. 

Here's what we do (copied from our website):Screening ApplicantsPoor screening often results in lost rents, legal & court costs and costly damages.Screening of applicants includes the following:Full credit/eviction/criminal checks are done, instead of score-driven reportsVerification of rental history includes validation of actual property owner to avoid fraudVerification of stability of income & employment via YTD paystubs & W-2(s) (analyzed to confirm not fakes)Income is calculated 3 ways to spot stability anomalies: pay period, YTD average and YTD + W-2 average.Debt-to-income analysis (more accurate than traditional 3x rent requirement)Bank statement to see if NSF issues, if deposits support income, prove MoveIn Funds, also analyzed for unusual transactions.Pet screeningHolding Fee required from approved applicants to make sure they are serious, before your property is marked pending.

What is good company to use for national criminal background check?

I was a prosecutor for seven years and a criminal defense attorney for two and that helps me with negotiation in real estate every single day.

I explained that since Marcus is a thief (he owes me money and stole some stuff from Bob), I am not obligated to host known criminals with known substance abuse problems in my duplex.

It would be a violation of the agreement and could result in losing my privileges to pull credit, screen for criminal and eviction, etc.