I have the following sentence:
Joe got everyone's attention and started to speak.
Should it be everyone's, everyones' or everyones?
As Robusto says, you should use everyone’s. Neither everyones’ nor everyones is a word.
Note that everyone is always singular and cannot be pluralized, which means everyones is incorrect. If everyones were a word, everyones’ would be the possessive form of everyones, but since everyones is incorrect, everyones’ is also incorrect.
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