Thanks everyone, I'm happy that "one's" is correct, especially as it's then consistent with "everyone's" which definitely looks right. Just looked a bit too much like "the dog and it's ball" to me. Context could be the Queen saying "One's handbag strap is chafing" vs "Ones handbag strap is chafing"
Could we have a context? I can imagine "ones" being correct occasionally- "these apples are riper than those ones" etc, but if it's one's grammar that's under discussion, the apostrophe is the only way to go, surely.
I was annoyed to see one's as a possessive in a book as I feel it's wrong. But later they then used everyone's. Which seems right. So should ones have an apostrophe. Apologies for errors in this am typing on phone and it's very frustrating.