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.
I think not. is used to disagree emphatically.
Sorry, but I don't have any references I can cite. Just my personal experience.
I think not is a formal way of saying no. It's often used in parody, faux and overly dramatic situations, to heighten the negation and introduce a sense of inadequacy, foolishness and insignificance.