I often hear people say things like :
"You don't like me using your telephone."
or
"You don't like me telling your age."

I was taught to use "my" in those instances, so I do, but have forgotten why. Or am I just being persnickety?

And then there are those pesky "Lying in the gutter I found my purse," things. Some can be quite hilarious but what are they called? And do you remember any from your English classes?
wow

Dangling participles.

me/my I have heard both. The me construction may be a readaption of you don't want me to tell your age. I guess I was taught that the my version was correct, too, but it doesn't seem to bother me all that much. It's like different than vs. different from or even different to. If rbarr (http://wordsmith.org/board/showthreaded.pl?Cat=&Board=words&Number=19694) were here she'd explain it all as dialect differences and choosing different models for a new construction, but she's not so we'll just have to fumble through for ourselves.

Just don't say "I don't like you.......and have that all the friend hears.

With no professional linguist afoot, this amateur asks y'all's indulgence:

We gotta do some parsing here. Or diagramming. Verbs that sometimes dress as nouns are called verbals. In the structure "you don't like my running," for example, the present participle running suddenly finds itself changed into a gerund (part of the verbal family), the direct object of the sentence, and as such, it needs an adjectival modifier - in this case, the possessive pronoun my.
Now let's take the verbal and make it transitive, as in the example Wow provided: "You don't like my running this equipment." Equipment becomes the direct object of running, which, while it hasn't forgotten its verb roots and functions as one within the expanded set ("running this equipment"), it still becomes part of the direct object of the entire sentence, needing "my" to modify it in the same way.

I am not certain of all the terminology; it's been years and years. But I figure if I don't know it, a lot of y'all might not either - this was my best shot at explaining this structure.

Her leaving misplaced and dangling modifiers for another time will probably be appreciated by kindred spirits who find over-long posts as tedious as she does

Undestandable explanation AnnaS better than the teachers I had, way back in the Dark Ages.... thanks
And for a short post, what's a favorite, funny "ing" saying.
wow

Fowler called the construction with an object rather than a possessive followed by the -ing form a fused participle. Generally it's better to use the possessive form of the pronoun but sometimes there is a difference in meaning between the two. Consider the sentence I like Alan/Alan's cooking in my kitchen. With "Alan", I think the liking is aimed at an oft-repeated activity, while with "Alan's" I think the speaker is talking about a specific occasion or series of occasions.

Some words resist the possessive in this context. Her meaning being clear did not make much difference. rather than Her meaning's being clear did not make much difference.

Bingley

Quick side note: As we've discussed in some other thread, it was a pity that Latin didn't survive as a required course. Slightly less archaic a lamentation: I wish they'd kept teaching these stinkin' kids how to parse a sentence! [Feeling a bit fogey today, I guess...]