Who’s is a contraction of who is or who has. Whose is the possessive form of who or which. Think of it this way: If you were to replace it with who is or who has, would its meaning change? If no, you want who’s. If yes, you want whose.
Here are a few examples of the words used correctly:
Celebrity birthdays: Who’s another year older Jan. 11? [OC Register]
Who’s Renting What on Netflix? [World’s Strangest]
Whose ass should I kick at ping-pong? [Warming Glow]
I am convinced this really is an idea whose time has come. [FDL]
Whose and inanimate objects
As in that last example above, whose—unlike who or who’s—may apply to inanimate objects or other non-person entities. For example, while you wouldn’t say, “The book, who is 500 pages, was released in 1923,” you could say, “The book, whose 500 pages fly by, was released in 1923.” This odd tendency arises out of the fact that there is no equivalent word in English that’s strictly for inanimate objects.