Should a 17-year-old student who lives at home file taxes on his own or with his parents?

Under tax law, as a student living at home, you are your parents' dependent for income tax purposes regardless of your income.

Therefore, you are their dependent and they will get that exemption. You, however, can file your tax return and, assuming you have earned income (from a job) that you had income taxes withheld on, you will receive a refund. If you made less than the standard deduction for a single person, you will get all of your taxes withheld refunded to you.

You must file a return if you had over just the minimum amount of income, regardless of your age. Read the discussions on that topic - who must file - here or at IRS.gov.

Regardless of it you must, generally, especially if you have a very small income, even if it isn't required for you to file, it will be beneficial, so you should. (And obviously if you worked and received a W-2 so you had tax or such withheld, you want to file to get the refund!)

If you are claimed as a dependent by someone else, you cannot claim the exemption for yourself too. And in fact, then the others (who presumably have a higher tax bracket than you), will probably have to pay (or you would have to pay) at the higher bracket.