It is important that you understand the Federal Fair Housing Act in terms of screening potential tenants. You cannot, for example, exclude a family simply because they have children.
It is important to set your criteria regarding what you require in an applicant in terms of credit scores, income, past landlord recommendations, etc.. and work from there.
There are wide variations among the states in terms of specifics that you cannot exclude a tenant for. For example, the newest issue in some states are "comfort animals", not service animals (such as seeing eye dogs or animals for the hearing impaired) but pets that owners claim they must have to help them deal with depression. There have been a (luckily) few cases where animal "hoarders" (people who keep large numbers of pets) claim that they must have ALL their animals to give them comfort, even if this large number are a source of damage to the rental unit.
Gail