What is the possessive of "you guys"?

Solution 1:

Regardless of the various permutations being thrown around these days, the correct possessive of you guys is you guys’, spoken or written, and I quote (emphasis mine):

To form the possessive case of a plural noun ending in s, add only the apostrophe.

EXAMPLES

• highways’ intersection
• the beetles’ legs

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^{John E. Warriner, Warriner's English Grammar and Composition (Orlando, FL: Harcourt Brace Jovanovich, 1986), 243}

Since you guys is a plural noun, it is subject to the above rule, which has no exception, as far as I know.

Indeed, you guys's is not an uncommon colloqiualism (I have even used it myself on occasion) but it is grammatically incorrect, and thus has no place in standard, proper or formal contexts, whether spoken or written.

Solution 2:

You guys' is correct written, pronounced you guys's. (Dropping the s on the written version because guys is a plural, not because I think it should always be dropped if the word ends in an s.) You rather than your because you guys is a unit and the possessive case doesn't need to be commuted across each element of it.