"Don't got" — how common is it in American usage?

In standard English, I think "don't got" has the same register as "ain't". It's definitely not correct, but very common. Avoid it in any high register situations where standard English is used, e.g. an interview, a presentation, a company report.

I sometimes will say "ain't", e.g. "that ain't worth it" in informal contexts but I don't think I would ever say "don't got".

I'd say the expression I don't got... is generally rare in spoken AmE. I'd think he/she/it don't is even more uncommon in the general population.

However, there are probably geographic or socioeconomic regions where both expressions are common. A COCA query yielded zero hits for don't got and 13 for don't have.

A final observation: If you really want to say I don't got... then you probably want to add a double negative. (ie I don't got no money.) I am sure there are lots of hicks who would say this daily.

American English doesn't have quite the divergent dialect issue that British English has, but it most certianly does have its own dialects.

What you generally hear from folks on TV is what is called Standard American English. "Don't got" would indeed be fairly unusual in that dialect, and would fall oddly on a SAE listener's ears.

However, there are other dialects where that is quite common, most notably African American Vernacular English, along with those spoken by many poor rural white folks. Features of AAVE are often found on USA TV shows (even coming from non-African Americans) to indicate that the speaker comes from a rough urban environment.

Here's what wikipedia says about AAVE and negation:

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Negatives are formed differently from standard American English:

• Use of ain't as a general negative indicator. As in other dialects, it can be used where Standard English would use am not, isn't, aren't, haven't and hasn't. However, in marked contrast to other varieties of English in the U.S., some speakers of AAVE also use ain't instead of don't, doesn't, or didn't (e.g., I ain't know that). Ain't had its origins in common English, but became increasingly stigmatized since the 19th century. See also amn't.
• Negative concord, popularly called "double negation", as in I didn't go nowhere; if the sentence is negative, all negatable forms are negated. This contrasts with Standard English, where a double negative is considered incorrect to mean anything other than a positive (although this wasn't always so; see double negative). There is also "triple" or "multiple negation", as in the phrase I don't know nothing about no one no more (in Standard English "I don't know anything about anyone anymore").
• In a negative construction, an indefinite pronoun such as nobody or nothing can be inverted with the negative verb particle for emphasis (e.g. Don't nobody know the answer, Ain't nothin' goin' on.)

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Given the skewed demographics of the US prison population, I think you could expect to hear AAVE or some poor rural dialect spoken in a prison scene, and would not expect to hear SAE.

If your question is strictly how common it is used in daily conversation, I can tell you that it is very common, at least in the southern US. As a high-school English teacher in this region, I can tell you that my students use it all the time, even in very formal writing. They're so used to using it, even my College Prep. students have trouble remembering to use a more proper form in academic writing. It's a huge headache for me!

I've heard it more often when people are negating things. E.g.

— We've got history class next.
— No we don't!

The correct negative answer to, "We've got history class next" is "No we haven't." The only correct usage of 'don't' in such an answer would occur if the question was, "Do we have history class next?"

'Ain't' is just a slang way of saying 'haven't' so the positive of "ain't got" would be "have got". "Do got" and "don't got" are incorrect in English, though they appear to be acceptable in American.