An example would be "It flew over everyone's heads", or "It flew over everyone's head".
What would be correct in this case and why?
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The answers to the question Is “everyone” singular or plural? focus on how, when "everyone" is used as the subject of a clause, we use the singular form of the associated verb (for example, we say "everyone was", not "everyone were").
Despite this, it is possible for at least some speakers to use plural pronouns and nouns alongside "everyone", because it may be interpreted as referring to a plural notion. Edwin Ashworth left a link in the comments to a related question about the use of plural or singular nouns in sentences with plural subjects: “They're using a cell phone” vs. “They're using cell phones”.
The Corpus of Contemporary American English (COCA) gives a frequency of 23 to "everyone's head", vs. 13 for "everyone's heads". The examples of "everyone's heads" were all from the spoken, magazine, fiction, and news categories; one of the examples of "everyone's head" was from the academic category. I would say that this establishes that plural nouns are used after "everyone's", although this is not as common and may not be as formal as using a singular noun.
As mpharries' answer indicates, some people might think of the use of a plural noun in this context as technically "incorrect". It seems the sort of thing that some copy-editors might feel strongly about, like "that" vs. "which" or so-called "singular they". In the wrong context (or perhaps for some people, in almost any context), it might also suggest nonsensical, silly imagery (for example, that you are talking about people who have more than one head each). So I wouldn't recommend going out of your way to use a plural noun in this context if a singular noun works. But sometimes, it seems to me like a singular noun would not in fact work (or at least, doesn't work as well).
Here are some examples I found on Google Books:
All in sync, everyone's heads readjusted so that they were still staring at him
(Stories of Earth: WWIII, by Dimitrious Charles)
I think this would require restructuring the sentence, not just replacing "heads" with "head", to avoid plural use. You see that later on in this sentence, the plural pronoun "they" is used to refer back to "everyone". Because of this, I think it would sound awkward to have "everyone's [singular noun]" in the first part of the sentence."Everyone's head readjusted so that he was still staring at him" is completely impractical and unnatural; you could do something like "All of the people's heads readjusted..." but I think "everyone's" sounds better than "all of the people's".
The idea is to cut off the tops of everyone's heads (again, not actually!) so that hats can be drawn on top.
(Caffeine for the Creative Team: 200 Exercises to Inspire Group Innovation, by Stefan Murnaw and Wendy Lee Oldfield)
While you could rephrase this as "cut off the top of everyone's head, so that a hat can be drawn on top", that restructuring makes it sound more like you should make several cuts. The sentence as written does a better job of implying that you are making a single cut, on a single piece of paper, that goes through all of the heads.
Like an affectionate mother, Karachi spread her torn, wornout canopy over everyone's heads.
(Mirages of the Mind, by Mushtaq Ahmed Yousufi)
This is similar to the previous example. I think the use of "heads" creates more of an implication that the canopy is over all the heads at once, while "spread her canopy over everyone's head" sounds a bit more like she spread her canopy over one person's head, then over another person's head, then another, one at a time. These implications are not absolute by any means (for example, the plural can be used for one-by-one actions: a sentence like "she touched their heads" doesn't necessarily say she touched all of them at once). However, that's the sense I get from it.
Suddenly credit scores appeared above everyone's heads, color-coded from green to red for severity.
(Freedom (TM), by Daniel Suarez)
I suppose it would be possible to replace this with "a credit score appeared over everyone's head, color-coded from green to red to severity", but the "from green to red" part sounds better to me after a plural noun than it does after a singular noun, and starting with "a credit score appeared" sets up a bit of a "garden path" where the reader doesn't learn until later on in the sentence how many credit scores appeared.
To address your specific example ("It flew over everyone's head[s]"): unfortunately, I haven't analyzed any data sufficiently to tell you which option would be more commonly used in a context like this, but to me it seems like either "head" or "heads" is possible in this context.
Here is a very similar example with the plural from Google Books:
At her command a silver bird lifted off from the deck, wobbled, and swooped over everyone's heads, dancing through the air.
(Everyone's Island, by Kris Schnee)
As in the examples I discuss above, I think the plural does serve somewhat to emphasize that whatever you're talking about flew over all the heads at once, not over each head one at at time. But since the latter intepretation is not expected anyway, I don't think the use of the plural affects the interpretation of the sentence very much, so I would be inclined to suggest that you use the singular "head".
Everyone is a collective singular pronoun.
Collective pronouns, referring to nonspecific persons or items, should be treated as singular though they have a plural implication or meaning.
Everyone takes a singular verb. But it is most often notionally plural. And plenty of native speakers are using everyone's heads and everybody's heads. However, there are a greater number of instances of everyone's head. (If I could figure out how to link to an Ngram that doesn't disappear when you link to it, I would.) So except on the most rigorous/prescriptivist test, it is safe to go with heads when you want to stress the multiplicity of heads.
Superman didn't fly just over Sue's, Joan's and Lois's heads, he flew over everyone's heads.
It is correct to say 'everyone's head' as each one has one head.
But, colloquially, it is the other choice that is often used :
Off with everyone’s heads!
The question, as asked, indicates an object in transit, not audience participation.