I agree 100% with zxc that in American speech, "I have got a guitar," and "I have a guitar" are equally common and interchangeable, and I think that most Americans today would not consider "I have got..." to be a "Britishism."

However, NB: the contracted form "I've a guitar" is not commonly heard in American speech -- one should either use the non-contracted form "I have," or else follow the contraction "I've" with "got." And this is true for the other personal pronouns, too:

1. They have a lot of apples. (GOOD)
2. They have got a lot of apples. (GOOD)
3. They've got a lot of apples. (GOOD)
4. They got a lotta apples. (CAUTION...)
5. They've a lot of apples. (BAD!)

The fifth construction (contracting "have" but without "got") might be seen in older poetry, music, and literature, but otherwise it is rare and uncolloquial in US English -- it's not grammatically wrong, but it's sounds strangely old-fashioned. If I heard someone say "I've ten dollars in my wallet," I would think: Гмммм, он либо англичанин, либо забытый советский шпион, которому не передавали, что Холодная Война закончилась, либо мартиянин -- а во всяком случае, не настоящий пиндос!

The fourth construction (dropping the contracted form of "to have" and just saying "got") is commonly heard, especially in youth and African-American speech, but tends to sound either slang-y or uneducated.

The first three constructions are all equally correct, but I would probably use #1 in formal writing and #3 in my normal conversational speaking.