I wonder if anyone can help me resolve a discussion my girlfriend and I are having please. Which of the following would be better English -

Mine & Adam's ball

Adam's & my ball

Personally, I would probably use either of those and not think anything of it, but I am told that I shouldn't use the first one as it is not proper English.

Anyway, any help would be appreciated, thanks in advance.
Hello, Tchrin, welcome to the Forums!
I personally wouldn't choose any of the above, I'd say "my and Adam's ball"
But then one is supposed to say the name of the other person first, rigth? [:^)]
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It would be more polite, but I don't think it would do as far as intelligibility/and grammar/ is concerned. This looks more like children's speach. There are many possibilities, and you have to consider whether the ball broke a window or not. Emotion: smile

Two children (I & Adam) play ball. An adult comes.
Adult: "What a beautiful ball! whose is it?"
Child: "This is my -and Adam's (in undertone) - ball"

Two children (I & Adam) play ball; the ball breaks a window; an adult comes:
Adult: "Whose ball is this?"
I: "This is Adam's ball! - (in undertone) and mine"
Adam: "It's his ball! - (in undertone) and mine"
Thanks for the welcome PieanneEmotion: smile.

Miche, that is my girlfriend's argument, that other people's names should come first.
Grammatically, the possessive case should come directly before the noun it relates to, otherwise you're bound to lose the thread...
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Mine & Adam's ball

Adam's & my ball

Personally, I would find both of these rather awkward and would probably say "the ball is mine and Adam's," or "the ball is Adam's and mine" or "the ball belongs to Adam and me."