This snowclone originated with the 1993 California Milk Board ad in which a collector of the very paraphernalia of a famous duel attempts to answer the contest question “who shot Alexander Hamilton?” with a mouthful of peanut butter sandwich. He cannot make himself understood, and finds he has run out of the milk which would help him wash that mouthful down, and so loses the contest. The ad ends with the words “got milk?”
As I recall, the ad was very popular, and was part of a set of similarly-themed ads in which the protagonist finds himself alone with a mouthful of something sticky and no milk. Its ubiquity is what I think helped snowcloneize the phrase, since “got X?” isn’t a particularly idiomatic construction.
This snowclone is used in situations where someone is trying to sell X, or it is presumed that X is something everyone needs or wants.
Instances of this snowclone are tricky to track down with snowclone.pl or Google, since searches on “got X?” or “got *?” are a little too permissive. Variations I have seen on bumper stickers and in my daily life include:
got sand?
got islets? [I want a t-shirt with this on it]
got rice?
got root?
got aloha?
got subluxation?
X does not need to be a one-syllable word, as the last example illustrates. It does need to be a noun, but the noun type is not particularly limited, as far as I can tell, though of course it does need to be something that can be referred to without an article (a(n)/the). (I.e., mass nouns, except I’m not sure that aloha and subluxation are considered mass nouns.)
More modern “got milk?” ads can be seen here.
The Snowclones Database 
In other words, it’s typically a mass noun. Plurals seem to me (impressionistically) to be much rarer. As for “root”, it’s a special case meaning “super-user privileges”, and thus a hidden plural.
I think this snowclone is now often used in the opposite context, to indicate something no one needs or wants, e.g. “Got lice?” or “Got steroids?”
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In middle school, I remember a classmate who had a t-shirt from Joe’s Crab Shack that said, “Got crabs?”
Yeah, he was eventually banned from wearing it.
No way, that was the first? I’m sure I saw plenty of “got milk?” commercials before the guy trying to say “Aaron Burr”. Though the exact order I saw them in is a bit hazy. There was a whole series of such commercials; I remember one of them was a fake story explaining the invention of the name “Oreo” for the cookie, because some guy had just stuffed himself with them and tried to say “I don’t know” when someone asked him what they should call it. But if that the Aaron Burr one was the first, then I simply didn’t see it until long, long after it was first broadcast. That or I’m confusing the other ones with “milk: it does a body good” which was in the 80s and early 90s. And then I remember the Sat AM cartoon Project GeeKeR doing an inane reference to it – the title character, who was a shapeshifter, and was in the form of a dinosaur, and caught a girl who had just fallen from a great height, and there was an awkward pause, which ended when he said “got milk?” and she screamed in terror. They used a lot of snowclones in that cartoon. Like “An enemy is just a friend who’s trying to kill you” and “trapped like a cabbage in a cabbage trap!” Someone involved in this website might make a little category for it, it’s probably a goldmine.