May i have everyone's attention,. Should it be everyone's, everyones' or everyones? What would be correct in this case and why?
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Your deduction, that not everybody's cup of tea means not to everyone's liking, is absolutely correct. The possessive of everyone is everyone's, in the same way the possessive of everybody is everybody's. In fact, to catch someone's eye is an idiom, not eyes. the question is not about everyone being singular or plural.
Unless of course you’re a buddhist or a cat, in which case everyone has multiple lives and both would be correct, depending on context.
Everyone who is coming will receive a gift. Are the words everyone and everybody singular or plural? An example would be it flew over everyone's heads, or it flew over everyone's head. There is a comprehensive article on the topic on grammar girl:
I have the following sentence: The indefinite pronoun 'everyone' is a singular pronoun that takes a singular verb.examples: The indefinite pronoun 'everyone' is a singular pronoun. The possessive form is everyone's (the form everyones' is incorrect).example:

Joe got everyone's attention and started to speak.
Would the noun following everyone's be plural? The expression commonly used in both the affirmative and negative. There are actually two issues concerning this topic: In my experience, the spelling checkers get confused by contractions.

