I've recently been told that "I don't think so" is, in the U.S.A., a southernism, whereas "I think not" is considered more acceptable everywhere else. Is this true?
Q: Is your wrist broken?
A: I don't think so. / I think not.
I think not! In British English, I don't think so. and I think not. are both used, although they have different nuances.
I don't think so. is more common, and shows a little diffidence or uncertainty.
A: Is it going to rain today?
B: I don't think so. I'd just wear a t-shirt if I were you.
I think not. is used to disagree emphatically.
A: You still owe me.
B: I think not! If I hear any more from you about this, you'll be hearing from my solicitor.
Sorry, but I don't have any references I can cite. Just my personal experience.