"my", "of me", "of mine" - when to use these possessive constructions
To express a possessive you can say "my X" or you can say "X of mine". The two constructions mean pretty much the same thing. No, "of mine" is not archaic, it's routinely used. But it is far less common than the "my X" form.
Bob is my friend.
Bob is a friend of mine.
Both mean essentially the same thing.
It's not limited to "friend", though now that you mention it, that may be the most common word used there.
The red one is my chair.
The red one is a chair of mine.
That said, "X of mine" is mostly used when you want to indicate that something is one of many that you own or are associated with. If, say, you have many cars, you might point to one and say, "That is a car of mine." But if you only have one, you normally say, "That is my car." (If you have many, you could also say, "That is one of my cars.")
That is, it's almost always "an X of mine", not "the X of mine". I don't think I've ever heard someone say, "That is the car of mine" or "She is the mother of mine"; it's always "That is my car", "She is my mother", etc. "The X of mine" would be grammatically correct, but no one says it.
Arguably it should be perfectly good to say "of me", there's really no need for the word "mine". But that's just not what English-speakers say, it's always "of mine". The only time I can think of when we say "of me" is when "of" is not indicating possession but rather is being used to mean "about", like "That's the story of me." (Even then, you'd be more likely to say "of my life" or "of my job" or whatever.)
(There is the phrase, "You're not the boss of me", but I think that's deliberately incorrect for effect. If I was simply stating the fact that you are not my boss because I have been assigned to a different department or some such, I would say "You're not my boss." "You're not the boss of me" is something you yell when someone's trying to order you around and you do not recognize them as having any authority, like when your big brother tells you to shut up.)
Like any construct that is not normally used but that is grammatically correct, people sometimes say "X of mine" for emphasis or poetic effect. Like I wouldn't be surprised to see "oh wife of mine" in a love poem or a Valentine's card, but no one says that in normal conversation, it would always be "my wife".
An object can belong to me, but it would never be of me, unless you are a pirate, or you were trying to state that the object came from your body (i.e., a piece of me), which isn't truly a possessive phrase.
The two proper ways to show possession are "my object" or "object of mine" — the use of the latter is often used to describe friends, but it is not limited to such uses:
Sally borrowed a book of mine.
Either phrase is technically correct in any situation. So to answer your questions directly:
"My thing" and "thing of mine" are completely interchangeable, though there is a modern tendency to use "my".
-
No, not even close. "Of me" might be used colloquially, but it is not proper English, and "of mine" essentially means "which belongs to me". The word "of" is used to show possession, as in:
He is of German heritage.
Which means that he belongs to the heritage of Germans. The word "mine" just happens to be the proper form of the word, "I", to be used after the word "of".
I, me, my, and mine are used with different functions in a sentence — "I" is the nominative case of the pronoun, used as a subject or predicate nominative, "me" is the accusative case of the pronoun, used as an object (direct or indirect), "my" is the possessive adjective, and "mine" is the possessive prounoun.
I would say that my is more personal and direct, of mine is more general and indirect.
Bob is my friend is much more emphatic - Bob and I are friends.
Bob is a friend of mine means Bob is one of many acquaintances that you are friendly with.
If you were one of many contributors to eg an art exhibition, you could show people round saying 'That's one of mine, and that's one of mine' without it sounding repetitious. If you said 'That's my painting' every time it would sound a little abrupt and odd.
Neither of these are connected to of me. My and of mine are possessive and must have an object in the sentence. Me refers to yourself, your body, or your intellect, and can stand alone.
This is a photograph of mine. (This is one of several photographs that I own).
This is a photograph of me. (I am the person in the photograph)
It’s a double genitive, which can occur both with a possessive pronoun (of mine) and with an independent genitive (of Bob’s). It can sometimes be expressed by other constructions, such as one of my friends and one of Bob’s friends.
It typically begins with the indefinite article, but it can also begin with another determiner when the noun phrase is post-modified, as in That friend of yours who came to the party seems rather nice.
Just want to point out that there seem to be some other valid usages of "of me" besides the ones Jay presented. In particular, constructions that end with "out of me".
Scare the hell out of me
Don't make a fool out of me
As in Jay's examples, these do not indicate possession.