Both forms are possible and one form is generally no more formal or popular than the other. Did you have a particular usage in mind when you asked your question? e.g. he does not have any money = he hasn't any money. We don't say: "he has not any money" - this does not sound natural. If this doesn't answer your question, please ask again. Kind regards, Michael

I agree with Michael. He hasn't any money - is absolutely fine. Well, HE might not think it's fine!