Who's is a contraction (and that is all it ever is), meaning who is or who has.
Because contractions are not appropriate for academic writing you will not need who's in your assignments—you would always write who is or who has in full.

Tell me, ♫ who's that girl ...running around with you? ♪

♪ Who's gonna tell you when it's too late ♫♪

♫♪ Who's been sleeping in my bed? ♪

You will notice that Who is... and Who has... will often occur as part of a longer verb construction.

Who's been eating the cookie dough.


Whose is a possessive adjective, with the sense of enquiring about what things belong to someone or something.

Although we talk in terms of ownership, sometimes the sentence isn't about ownership in the traditional sense, but about responsibility for.

Whose car is that in the driveway?

I don't care whose idea it was, I just want that elephant off my property.

♫♪ Well, whose bed have your boots been under? ♩

Whose as a relative pronoun

Whose is also a relative pronoun
Remember our lesson on the difference between who and whom?
We found the following explanation at: Language Portal of Canada — they explain that who is the relative pronoun for when the person (in question) is the subject of the sentence, conversely whom is what we use when the person (in question) is the object of the sentence, and finally whose signals possession, or ownership, (or belonging to

The officer, whose duty it was to control the crowd, had over-reacted.

O thou, in whose presence, my soul takes delight. ♫♪

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