While there are multiple Google searches and free databases you can search on a potential tenant, most results are likely outdated. Some individual counties will keep up-to-date online criminal records that are free to access, but they seem to be few and far between. You can check with your county and surrounding counties to see if this service is available.
Your other option is to go to a
tenant screening service. My company, AdvantageTenant.com, provides tenant credit checks, criminal background checks, eviction reports, sex offender search, employment verification and other reports either individually or as a package. All pricing is on the site.
In all fairness, there are many other companies who do this. Feel free to check them out as well.
It seems that most landlords and property managers believe that the credit report is the most predictive tool for assessing a tenant, but they also run a criminal or eviction check to cover their liability.
Hope this helps.