When do you stop accepting applications for a rental property?

[Wade Penner]

This is my first time through the screening process, and I had everything dialed in to screen multiple applicants and choose the best based on a set of criteria.  However, I've now shown it a few times and there was one seemingly qualified applicant who submitted an application.  I'm currently going through the process of screening her - already ran background and credit checks, and I will be verifying rental history today.

Meanwhile I have other people emailing me about the ad that is still up.  When do I draw the line of picking this applicant versus continually evaluating more potential applicants?  Presumably if I leave the ad up then people will continue to email me.  There was nobody else applying when I showed it to her, so I made it sound like there was a good chance she would get it if all checked out on the application, otherwise I may have tailored my message differently (if there were others actively applying still).

p.s. for what it's worth she has an emotion support dog ... sounds like it's wise to be extra careful not to discriminate when screening.

Thanks!