Things I look for, and include in my listing: no evictions or major criminal background issues, take home income at least 3x monthly rent (this is personal requirement, and higher than the typical rent = 30% of gross).  

SEC 8 ONLY screens for criminal. 

And if they apply and pay the cost I run their credit and criminal. 

I have the income requirement on the application now, but I'm curious if other PMs list things such as minimum credit score, eviction-free period, and criminal check history requirements.

No criminal history (or whatever level you are comfortable with)Speak with past landlords, employers and have a background check completed. 

You are not liable if you unknowingly rent to a criminal. 

I think the Apartments.com one used to be Cozy and it seems like it has additional functionality (especially around maintenance requests)Second question: Anyone used the applications feature (credit score, background, eviction, criminal, etc) of either of those two sites?

Last I checked it was to illegal to discriminate on the grounds of an eviction and most criminal activity.  

On the other side of the political spectrum, it’s about the plight of our most vulnerable citizens — and how the city has essentially criminalized their very existence.

(Or more likely settle the suit) if you are negligent or criminal you would probably be personally liable anyway.