I feel your pain. Even I have to think about these, most of the time, and I think grammar all the time. (“Hey, I just got an email from my grandmother, and she really should have used a comma there. And this forum post has the wrong ‘there.’”)
Whose, who’s. It’s, its. Your/you’re. And, for good measure, we’ll throw in let’s/lets and they’re/there/their even though the first is not a possessive but a contraction.
Here’s a general rule of thumb. If it has an apostrophe, it’s a contraction, not a possessive. This is where the majority of people get confused, because we’re all taught in elementary school that if something ends in ‘s, it is a possessive. This is true for just about every noun you can think of, but it is not true of pronouns. Not a single pronoun follows the ‘s possessive rule.
(For those of you who just went “Holy Batman, I don’t remember what a pronoun is,” it’s a word that replaces a noun. I, you, we, he, she, they, it, etc.)
Every pronoun has a unique possessive form. My. Hers. His. Ours. Your. Their. Its. And, yes, whose. Whose is the possessive.
Who’s is a contraction. Who is. As in, “Who’s that staring in my window?” Whose, possessive. “Whose window is this that I am staring in?”
PRONOUN POSSESSIVES DO NOT HAVE APOSTROPHES. EVER.
It’s. Contraction for it is. “It’s a balmy 75 degrees out.” Its is the possessive. “The dog wags its tail (and then goes upstairs and eats the cat’s food just to spite me).”
You’re is a contraction for you are. “You’re getting jello on the couch.” Your is the possessive. “Come get your damn jello before it stains.”
Let’s is a contraction for let us. “Let’s go to the zoo and throw Dave in the tiger cage.” Lets is the present tense he/she/it conjugation of the word ‘let.’ “He lets the cat into the room.”
They’re is short for they are. “They’re coming over after dinner.” Their is the possessive. “Oh, they forgot their mittens.” There is a vague location-based term. “Well, just set them over there.”
So, if it is a pronoun possessive, remember, no apostrophe. And if you don’t think you can remember that, just look at it and see if there’s a contraction that easily comes to mind instead. If so, no apostrophe.