Native North American speaker here. It's fairly common in certain British dialects to substitute "me" for "my" (Shiver me timbers) in informal speech.
My impression is that some speakers mix the two.
What are the descriptive rules for selecting between the two variants?
Specifically, I'm asking about usage of
A non-standard variant of my (particularly in British dialects) is me. (This may have its origins in the fact that in Middle English my before a consonant was pronounced [mi:], like modern English me, (while me was [me:], similar to modern may) and this was shortened to [mi] or [mɪ], as the pronouns he and we are nowadays; [hi wɒz] he was; versus [ɪt wɒz hi:] it was he. As this vowel was short, it was not subject to the Great Vowel Shift, and so emerged in modern English unchanged.)"
As an example, here is an excerpt from the character Daisy in Downton Abbey:
Someone walked over me grave.
Someone please supply a video clip of this speech pattern. I'm sure it can be found, but I can't.