From the HUD website:"In the Sale and Rental of Housing: No one may take any of the following actions based on race, color, national origin, religion, sex, familial status or handicap: Refuse to rent or sell housingRefuse to negotiate for housingMake housing unavailableDeny a dwellingSet different terms, conditions or privileges for sale or rental of a dwellingProvide different housing services or facilitiesFalsely deny that housing is available for inspection, sale, or rentalFor profit, persuade owners to sell or rent (blockbusting) orDeny anyone access to or membership in a facility or service (such as a multiple listing service) related to the sale or rental of housing."

Information on the tenants Look up the crime rate in that area Look up nearby sex offenders that are registered - it can be good to know how many and how far away they are in case you rent to a family, etc.

It is legal to collect advance payments here but maybe they have a problem you have not discovered yet like being a registered sex offender or something and want to lock up a long term before it is discovered.

You aren't allowed to consider criminal history except for sex crimes as part of a background check,  and you must accept the first qualified application according to your (legal) rental criteria.  

57 sex offenders within 0.5 miles of the property.

Two National Eviction Reports Includes: Sex Offender, Terrorist, Most Wanted Criminal 3.

We do not discriminate against persons because of race, color, religion, sex, handicap, familial status, national origin, age, sexual orientation, or gender identity.

Well, in CA anyway, we're required to give the Megan's Law database info in our leases, so tenants are aware of it, and can look up sex offenders on their own. 

I found one of my potential tenants was apparently dating a registered sex offender.

What if that felony that they have just been convicted of requires them to register as a sex offender?