As others have pointed out, we cannot comment on the legality of such discrimination without knowing the location. However, even if it is legal, it seems like a terrible idea. Much of thisthe following has been covered in the comments, but I just want to summarise what I see as some major false premises stated or implied in your question:
Single people have more "free time" available than those in relationships
Being single does not exclude someone from having other hobbies and interests, or from spending time on the dating scene. And indeed, it may mean that certain day-to-day tasks are more time consuming as there isn't someone else to help do the cooking, cleaning, etc. etc.
Single people are more "driven" in their work
I don't get where this comes from. I guess it's true in a few cases that some people are single because they are obsessed by their work and don't have time for relationships, but this is by no means universal. Also, someone in a relationship might be driven by the feeling that they need to support their partner/kids. Importantly, relationships can provide emotional support and stability that, if lacking, might in some cases lead to a loss of motivation.
Spending more time working is inherently more productive
I think this has been fairly well debunked. Rest and variety is very important to remaining productive.
Being in a relationship makes one a less well-rounded person
?? Why would this be the case?
Accepting a student who is currently single will ensure that they remain relationship-free for the next 4 or more years.
Is the student going to be dropped if they start dating someone 6 months into their studies?
I recognise that in a technical sense this does not answer the question, but I am trying to point out that the premise behind the question is deeply flawed. This is also too long to include as a comment. However, if you feel it is inappropriate, then down-vote and I will consider deleting.