When asking if someone is not married or in a romantic relationship, you can ask, "Are you single?".
That I know.
Now, can you also say any of these to mean the same thing?
(1) Are you a single?
(2) Are you a single guy/girl?
(3) Are you a single person/one?
I don't think #2 is unimaginable. Given the popularity of the book Sex and the Single Girl, I have to think an exchange like this probably happened in bars in the 1960s and 1970s:
- Someone as pretty as you must already have someone special in her life. Still, a fellow can hope. So...are you a "single girl", then?
- Well, actually, I've just broken things off with my boyfriend.
- How lucky for me! This night is looking better and better.
People say things like "So, you're a swinging bachelor now?" after a man breaks up with his partner, so I can imagine someone playfully asking "Are you a single girl?" in the same spirit: evoking the type of the "single girl", i.e. a romantically/sexually liberated young woman with her options open.
That said, #1 is still the default choice by far -- though as a direct question, it can also read as rather formal. At least to me, "Are you single?" makes me think of filling out paperwork during meetings with an accountant, or the person in charge of health insurance. In informal settings, people often find another way of asking the same question:
Are you seeing someone/anyone?
Do you have a boyfriend/girlfriend?
[Is there] Anyone special in your life these days?
Are you in a relationship?
On the other hand, asking about a third party -- "Is he/she single?" -- carries none of the same formality (again, at least to my ears). Something about directly asking a person "Are you single?", though, strikes me as oddly...clinical.