What quotes from Shakespeare's play Hamlet that support the idea that Hamlet was mad?

There are numerous incidents and quotes in Shakespeare’s Hamlet to which we can point as potential proof that Hamlet is mad. In the first act, after his father’s ghost tells Hamlet he has been murdered and asks him to avenge him, Hamlet returns to Horatio and Marcellus, who ask how things went with the ghost. Hamlet becomes defensive, claiming they will spread any news he shares, and despite the fact they say they will not, Hamlet tells them to go about their own business, and he will go pray. They respond, “These are but wild and whirling words, my lord” (1.5.127-134). They are essentially telling Hamlet he is not making sense.

Later, Ophelia, the woman Hamlet (supposedly) loves, tells Polonius that Hamlet badly frightened her when he

with his doublet all unbraced;
No hat upon his head; his stockings foul'd,
Ungarter'd, and down-gyved to his ancle;
Pale as his shirt; his knees knocking each other;
And with a look so piteous in purport
As if he had been loosed out of hell
To speak of horrors (2.1.1035-1041).

Polonius brushes it off, asking if perhaps it is madness prompted by the depth of Hamlet’s love for Ophelia that has made him appear so, but Ophelia goes on:

He took me by the wrist and held me hard; 
Then goes he to the length of all his arm, 
And, with his other hand thus o'er his brow, 
He falls to such perusal of my face 
As he would draw it. Long stay'd he so.
At last, a little shaking of mine arm, 
And thrice his head thus waving up and down, 
He rais'd a sigh so piteous and profound 
As it did seem to shatter all his bulk 
And end his being. That done, he lets me go, 
And with his head over his shoulder turn'd 
He seem'd to find his way without his eyes, 
For out o' doors he went without their help 
And to the last bended their light on me. (2.1.1046-1059)

Basically she says that Hamlet grabbed her wrist, touched her face as if he were going to paint it, then starting bobbing his head up and down like he had gone crazy, then left the room without taking his eyes from her face. Polonius determines then that Hamlet has gone crazy because he has not been able to see or be near Ophelia.

Later still, Hamlet goes to visit his mother, hears someone, who turns out to be Polonius, behind the curtains, and kills him. His father’s ghost reappears, but significantly, Hamlet is the only one who can see him. To this point, others have also always seen the ghost, but now, when Hamlet alone can see and hear him, his mother declares that he is mad (3.4.2500-2540).

Last, when facing Laertes in a duel over Polonius’s murder, Hamlet makes his own declaration, stating that he killed Polonius in a fit of madness:

Give me your pardon, sir. I have done you wrong; 
But pardon't, as you are a gentleman. 
This presence knows,
And you must needs have heard, how I am punish'd 
With sore distraction. What I have done 
That might your nature, honour, and exception 
Roughly awake, I here proclaim was madness. (5.2.3863-3869)

There are also other possible indications of Hamlet’s insanity, such as his violent mood swings, especially with his mother, and his irrational behaviors, such as boarding a pirate ship with no backup and killing his friends Rosencrantz and Guildenstern for no particular reason. The problem is, however, that Hamlet tells us at the beginning of the play that he intends to act mad:

Here, as before, never, so help you mercy, 
How strange or odd soe'er I bear myself 
(As I perchance hereafter shall think meet 
To put an antic disposition on),
That you, at such times seeing me, never shall, 
With arms encumb'red thus, or this head-shake, 
Or by pronouncing of some doubtful phrase, 
As 'Well, well, we know,' or 'We could, an if we would,' 
Or 'If we list to speak,' or 'There be, an if they might,' 
Or such ambiguous giving out, to note 
That you know aught of me- (1.5.922-932)

Then, when he sees the ghost in his mother’s chamber, and she tells him he is mad, he assures her that he “essentially [is] not in madness, [b]ut mad in craft” (3.4.2592-2593).  Hamlet also tells Guildenstern that he is faking madness:

Hamlet: [M]y uncle-father and aunt-mother are deceived

Guildenstern: In what, my dear lord.

Hamlet: I am but mad north-north-west. When the wind is southerly I know a hawk from a handsaw. (2.2.1457-1461)

So although there are quotes in this play that seem to indicate Hamlet is truly mad, there are equally as many that call into question the veracity of his madness. In the end, it is perhaps more his behaviors that allow for an analysis of his sanity.