All three are correct, but the first is dated and not really used nowadays.

There is an excellent discussion here: https://www.italki.com/question/484604 [excerpt from an excellent answer by Su.Ki., a respected italki member] Remember this: 'Don't have' is always right. 'Haven't got' is sometimes, but not always right. 'Haven't' is usually wrong or very unnatural. You can forget about it.

In US English, you’d never use the first one. It would be understood, probably, but sounds strange. The other two are fine.

I haven't had a car. I don't have a car. I haven't gotten a car.

I haven't a car. Incorrect. I don't have a car. Correct and probably the most natural way of saying this. I haven't got a car. Correct