Is either one of these correct?
"I want to compliment someone on their hair", "I want to compliment someone about their hair."
"I like what you've done to your hair", "I like what you've done with your hair".
on their hair" is okay. about their hair" is not used in American English. to your hair" is okay.
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"...on their hair" is okay. "..about their hair" is not used in American English.
"...to your hair" is okay. "...with your hair" is insulting-sounding, like her hair was a mess and she finally did something about it.