What exaples of madness does Hamlet in shakespeare exhibit?

A very important scene to look at is Act 2 Scene 1 from Ophelia's entrance. Also look at the conversations between Hamlet and Polonius (Act 2 Scene 2) and the way his speech has changed. However, there are arguments to say that Hamlet shows no signs of madness and is in fact acting the whole time. This is made most obvious from the quote "I am but mad north-north-west: when the wind is southerly I know a hawk from a handsaw." and from his soliloquay at the end of Act 2 Scene 2 ("Oh what a rogue and peasant slave I am...")