Unless the individual is a convicted felon, they have a constitutional right to bear arms.

I just posted this story in an other forum on biggerpockets. a guy calls interested in one of our apts, we send him to our web site where all of our qualifications are listed. he makes an appointment to view the apt very clean cut guy brings his application with the $35 bucks and pulls up in an very well kept older BMW makes $50k a year and single I want this guy as a tenant. we do the background check and it is one of those OMG moments 14 pages of felony convictions and a few evictions 32 felony's in total. so bad I was thinking omg i was alone in the apt with this guy. so don't worry about the how heavy people are just be diligent with qualifications   

It's just a list of who had an action filed.This would be the equivalent of denying someone who was arrested, regardless of whether or not they were convicted. 

Multiple convictions for driving without a license.

Tough for a person to qualify for Sec 8 with a criminal record, depends, but drug convictions are an automatic rejection, It is possible that an applicant may have some minor offense, but if your qualifying standard for everyone is no convictions, you are not forced to accept that applicant just because they receive Sec. 8.

Something else to consider is a recent criminal conviction can cause deportation. 

If yes, how many times ___ 4) Have you ever been convicted of a felony?

The tenant can violate their entitlement agreement (or whatever it's called) for personal reasons - like getting behind on child support, drug use, or having a guest who is a convicted felon on the property, but if the actual property does not pass - even if it fails because of a condition the tenant caused, my experience has been that it's going to fall on the homeowner. 

Now, that's not having been convicted or sued for discrimination, that's if they think you did it before.

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