Are you rejecting someone because they are a danger to your investment or to others?

Credit reports are a extremely important factor but by no means the total picture.The purpose of the screening process is to identify reasons to reject a applicant.

Thankfully, the large PM co's reject them for no FICO.

Find out the details and if it in any way places future risk on you as a landlord you simply reject them.Any judgement regarding a previous landlord is a automatic rejection.

Now the counter-offer is the offer and you can accept or reject it.So, when you verbally offered to renew the lease and he came back with - paint the apartment the way I want and I'll stay - that was a counter-offer. 

As a landlord I mitigate risk and decide on accepting or rejection based on all the data collected.

Either take it up front or reject them.

I reject anyone I do not want for any reason I choose and am always able to defend my decision but surprisingly have never had to.

@Dan Shelhamer, you must check your county/state/city laws because some areas indeed do consider it discrimination to purposely reject people just for being on Section 8.

If they fail your application process you have a little less obligation - although you may have a legal one if it is a credit-based rejection.