Which apostrophe is correct? (There are multiple dogs.)
One of the dogs' tails
One of the dog's tails
I believe it is the former but I'm not 100% certain. I found absolutely no guidance on this when I searched on the internet.
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The apostrophe follows the final s if that is the last letter of the owner's name. It is not "Janus's" faces but "Janus' " faces that look both ways. When saying this out loud it seems you have forgotten the possessive when you did not.
Compound nouns in such places get confusing. It is often better to rearrange things; "The tale of one of the dogs is wagging."
But, in requiring an answer to your question: "One of the dog's tails" is the correct usage in the references I can find. Look for compound noun possessive to see for yourself.