My landlords are very happy! I have rarely had a 'bad apple' get past all of that, not that it doesn't happen, it does, but it's rare.
Gone but an earlier landlord will. Some landlords ask the tenant to get background check in themselves from local police dept (highway patrol is better they pull the state not the municipality) - but if they just moved to the state nothing may show from previous state so some landlords will use a service that will check further (teants are asked for a nonrefundable application fee to pay for it). Most will ask for a credit report - they will typically accept the persons yearly free one or they will have the background company run it. Other lanlords use their gut and ask nothing more than rents and deposits
Traditionally, landlords have considered credit status and previous rental history along with employment to determine if a prospective tenant could afford the rent and established a track record of responsibility. More recently, some landlords have added criminal background checks to their screening process.