Is it permitted to not take Ph.D. students who have girlfriends/are married/don't come in weekends...?
A disclaimer:  I'm not an attorney.  My opinion is my own but I'll give you my sources.

Here in the US only, universities are probably not prohibited from considering otherwise protected status in admissions, c.f., Fisher v. University of Texas at Austin, which upheld racial preferences in admissions.

But under Title VII of the Civil Rights Act of 1964, as amended, once the question becomes one of employment (e.g., they work for you and get paid a stipend) you cannot discriminate based on any of the factors you cite.  You should also carefully avoid asking questions that suggest you're even thinking about it, c.f., "Federal Laws Prohibiting Job Discrimination Questions And Answers" and "Pre-Employment Inquiries and Marital Status or Number of Children" on the EEOC site.

Added: I agree with Anonymous. Even if you think you can get away with it, perhaps because it's not illegal in your country, I also think it would still be wrong.

Added:  If you are the victim of unlawful discrimination in the workplace, HR is not your friend. A priori, their objective is to make sure discrimination complaints don't happen. But after the fact, their objective is to circle the wagons and make sure a complaint dies without the institution having to pay a settlement or admit that any wrongdoing has ever happened.  A great outcome for HR is that you just quit and the problem goes away.  From your latest performance reviews and interviews with your co-workers, it's clear you were unhappy and have been for months and it's been showing up in your work.  So you quit.  Poof!  Problem solved.  Separately, they'll figure out what to do with the others involved.  If you think you may be a victim of unlawful discrimination, you should go first to your own attorney for competent legal advice, not HR.