I say that, but it’s halfway between “I’ve” and “I have.” Same when I say “I’ven’t.”
I rush it, so it sounds like “I’ve,” but, really, that’s what it is. Why’s it bug ya? 
“I haven’t.” My ESL teachers didn’t even teach us “I’ve,” much less “I’ve not.” That doesn’t mean it’s incorrect, but it is a dialectal variation that doesn’t get taught to us furriners; I don’t know whether it would be one of those things that native speakers do but ESL teachers cross off in the exam, or a case where mileage will vary.
I use “I’ve not”.
I grew up in Nevada, but spent the last 7 years overseas and about 75% of my friends are UK or Aussie natives.
Ooh, now you’re asking. Either seems perfectly fine and natural to me, and I’m sure I use both; though I suspect I tend to say “I haven’t” more often than “I’ve not”.
I have an idea that “I’ve not” may be more typical of Scottish English than English English, but I could well be mistaken.
I’m an Aussie and I have only heard the contraction “I’ve not” used by Brits because, I presume, in every other circumstance it can’t be used really.
“Haven’t we met before?”
“I haven’t got any idea where that quote is from.”
“We haven’t got them in that size.”
“I haven’t even started yet.”
etc.
“We’ve met before, haven’t we?”
“I’ve no idea where that quote is from.”
“We’ve not got them in that size.”
“I’ve not even started yet.”
British, and I would say “I haven’t”. To me “I’ve not” sounds like a northern English dialectal thing. Also “do you not?” instead of contracting it to “don’t you?”
Northern England, and I use both, but probably “I’ve not” more often than “I haven’t”.
“I haven’t.” “I’ve not” sounds all wrong in American English, although “I’ve never seen a Harry Potter movie” sounds fine.
But “I have never seen a Harry Potter movie” and “I have not seen this particular Harry Potter movie” clearly have two different meanings, don’t they?
Yes, of course they do, but my point is that “I’ve never…” is a common phrase in American English, whereas “I’ve not…” isn’t.
Nor.
That one: “Seen Harry Potter?”
Myself: “Nor.”
This is interesting.
Is it a choice you’ve adopted, something you you grew up hearing, or what? And, if you heard it growing up, where was that?
Or, is this my first whoooshie thingie?
I think it has to do with emphasizing “I” after others have commented.
Joe: Have you been to Thailand?
Mary: Yes, it is beautiful.
Paul: Oh yes, I love it.
Al: I have been there many times.
Me: I’ve not been there yet, but would love to go.
Just my gut feeling, as both are grammatically correct, but I think I use it mostly when differentiating myself from others who have chimed in before me.