Negation and in particular how negative contraction competes with verb contraction is a very large area of research. For example, The Longman grammar of spoken and written English devotes thirty pages to it, and there's some information in the appendix, too. Since there is so much detail there, and your question is about "I've not" vs. "I haven't", I'll answer that particular question.
Several posters suggested that "I've not" might be a Britishism. However, various researchers argued that with the verb "have" there is a very strong preference for negative contraction even in British English:
I haven't is much preferred to I've not. (The Longman student grammar of spoken and written English, 2002: 242)
They haven't finished is more common than They've not finished (Greenbaum 1992: 684 in The Oxford companion to the English language)
The Longman grammar of spoken and written English (1999) has Table A.8 in the appendix (on p. 1132). In that table, you can also see that "haven't" is "overwhelmingly the preferred choice" (p. 1131).
It's not entirely clear whether you're interested in the lexical or auxiliary "have". In any case, even in British English the lexical verb "have" sounds formal or old-fashioned (e.g. I haven't a brother). Corpus studies have shown that this use of "have" is pretty rare - the more usual being "I haven't got a brother" (BrE) or "I don't have a brother" (AmE and BrE).