Where are phrases such as "my one friend" used?

Very much conversational and typically at the start of a story:

My one friend had this cat named Hubcap and one day...

So also the kind of thing you might hear in a stand-up comedy routine. I would suggest it is perhaps more likely to turn up in certain dialects or modes of speaking, but I'm not sure which ones or where it originates. In most places you would be more likely to hear "one of my friends" or "my friend" instead.

Actually, when someone says "my one friend" its meaning is contextual. Consider the following:

"I'm 101 years old and my one friend just died."

In this case, it's likely the speaker truly means he had but a single friend left. For clarification, he might have said "my one remaining friend," but the listener would understand that he is now bereft of friends.

On the other hand, the nominal phrase "my one friend" introduces the notion that the speaker desires to emphasize specificity in some special context. It's similar in construction to "this one guy even asked me if I would give him some money after I asked to borrow five bucks" or "this one moron held the rabbit by its ears" or "this one lucky bastard hit his flush on the river and sucked out on me": it intensifies the focus on a particular subject, and alerts the listener that what follows is of special note.

As to your actual question, I don't think there is a regional flavor to this locution. I have heard it all over the U.S., but it's always an informal usage and I want to say it has a working-class feel to it, but that's probably not accurate to several decimal places. I think you are more likely to hear it said by a person who also says "she goes" to mean "she said," as in "I asked my one friend to stop texting my boyfriend a gazillion times a day, and she goes, 'Yeah, right, bitch. I'll text whoever the hell I want.'"