To give practical examples:
She does not like my jacket. -> my jacket is a noun phrase and object of "like"
*She does not like me jacket -> obviously wrong.
She does not like singing (singing is a gerund and object of "like")= She does not like the action of singing (the object of "like") (The context will tell you if she does not like to sing, or if she does not like to hear anyone who is singing.)
She does not like {my singing} (Verbal noun phrase and object of "like") = She does not like {the sound that I make when I sing.}
She does not like me singing = She does not like me when I sing (but she might like you if you sing.) Correctly, this would have a comma after "me". (It is also possible to understand this as "when she sings, she does not like me.")
In prescriptive terms, only "my singing" is correct, but so many people use "me singing" that "singing is understood as a post-positional adjective qualifying "me" = "She does not like the singing me."