Referring to a friend like: "the friend of mine" is a common way.
But what about referring to something else, like car, fork or plant i.e. the something of mine? Is there any restriction of using subjects only?
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(First off, it's much much much more natural to say "a friend of mine". I tried but failed to come up with a natural-sounding sentence that uses this phrase with a definite article.)
You can, grammatically speaking, use the "of mine" construction with pretty much any noun that you are capable of possessing.
That looks very much like a fork of mine.
We used a book of mine to settle the debate.
However, be aware that this construction always sounds a bit stilted, and it means something slightly different than the straightforward "my [noun]" form.
He drove a car of mine to the restaurant. [The car is one that I happen to own; I probably own others.]
He drove my car to the restaurant. [The car is probably my only vehicle, and it's the one I usually drive when I go someplace.]
If in doubt, stick to using "my [noun]" except in the set phrase "a friend of mine".
of mine is an old construct that means not only possession but also emotional attachment.
It's currently rarely used outside of "Friend of mine". You could say "House of mine", or "School of mine", about places where you spent memorable moments of your life, or "child of mine" about your dearest baby. Saying "Lady of mine" is a fancy way of introducing your girlfriend or wife.
If you use it in context of common items, it will have comedic effect. Imagine yourself cradling a fork to your heart tenderly, and whispering "This is a fork of mine" with tears in your eyes - that's the right context for this usage.
There are several usages where mine is preferred versus my:
In other cases, my and mine are almost equal.
Also, to expand @Martha's answer: "A friend of mine" may suggest that you may have more than one friend. Compare:
The first usage may suggest that you likely have several pens, while the second one suggests you sacrificed your only pen to someone.
This suggestion may vary in different dialects, however.