sure,
And take a bond of fate. Thou shalt not live,
That I may tell pale-hearted fear it lies,
And sleep in spite of thunder.

Macbeth is so obsessed with the witches' prophecies that he's starting to feel untouchable. Though this may not be a sign of madness, it's certainly a sign of dangerous arrogance and hubris. Macbeth's complacency about the threat of Macduff is misplaced indeed, as Macduff will be the man who finally kills him.

In act 5, scene 5, even as Macbeth approaches his doom, he's still firmly in the grip of self-delusion. This is especially dangerous as it's robbed him of the fear that he ought to have for his enemies:

I have almost forgot the taste of fears.

The time has been my senses would have cooled

To hear a night-shriek, and my fell of hair

Would at a dismal treatise rouse and stir

As life were in't. I have supped full with horrors.

Direness, familiar to my slaughterous thoughts

Cannot once start me.

Macbeth is so deluded that he feels himself protected by the witches' prophecy. Horrible things have become so familiar that they no longer startle him. For a man in his position, about to be attacked by his enemies, that's a very dangerous attitude indeed.

Macbeth's mind has become more unstable since killing Duncan and Banquo.

The fact that Macbeth is conjuring witches and asking them for advice shows a little bit on instability.  However, accepting that he is otherwise sane and there is just magic afoot, Macbeth is well into a downward spiral by the fourth act.  He is getting more and more paranoid, and his paranoia continues his murder spree.

Killing Banquo because he might have been suspicious (and because the witches said his sons, and not Macbeth’s, would be king) was just the first of Macbeth’s murders.  He seems to think that time itself is against him.

Time, thou anticipatest my dread exploits:
The flighty purpose never is o'ertook
Unless the deed go with it; from this moment
The very firstlings of my heart shall be
The firstlings of my hand. (Act 4, Scene 1)

These are the ravings of a madman.  Macbeth is basically saying that if he thinks it, he will do it.  He decides that Macduff is a threat, so he will kill Macduff.  He sends murderers to kill Macduff, and they kill his wife and son.  Lady Macbeth is worried that her husband has lost his mind.  Even though she does not know the details, she is afraid.  She confides in Ross, who tells her to have patience.  She responds that Macbeth is the one with no patience.

He had none:
His flight was madness: when our actions do not,
Our fears do make us traitors. (Act 4, Scene 2)

The fact that his wife, who planned the murder of King Duncan so meticulously, thinks that Macbeth is mad shows how far he has gone.  By the time Malcolm and his army are storming the castle, Macbeth’s wife is dead and he has lost his faith in himself.

Out, out, brief candle!
Life's but a walking shadow, a poor player
That struts and frets his hour upon the stage
And then is heard no more: it is a tale
Told by an idiot, full of sound and fury,
Signifying nothing. (Act 5, Scene 5)

Proof of the fact that Macbeth has lost his mind is his reaction to Macduff when the two meet in battle.  He was told two seemingly contradictory prophecies by the witches, that no man born of woman can hurt him, and that he should beware Macduff.  When Macduff tells him that he was “from his mother's womb/Untimely ripp'd” Macbeth loses all confidence that he is safe (Act 5, Scene 8).  It is actually pretty easy for Macduff to kill him then.

Macbeth was not all too sane to begin with.  It was very easy for him to be manipulated by the witches and his wife.  Soon he was killing left and right to protect a throne he wasn’t even sure he wanted.  Macbeth’s downfall seemed inevitable. He was destroyed by his own ambition.

Macbeth's mental deterioration begins when he kills King Duncan and continues as he gradually becomes a ruthless murderer, while Lady Macbeth's mental decline is the most conspicuous in Act 5 in the famous sleepwalking scene.

Firstly, let us focus on Macbeth's hallucinations. His ambition to murder Duncan so that he could become the king proves to have a very detrimental effect on him. Although he achieves his dream of becoming the ruler of Scotland, he loses his inner peace and stability for good. This is evident in the scenes when he appears to behold certain objects and apparitions, such as the invisible knife, the ghost of Banquo, etc. Here are some of the quotes that demonstrate Macbeth's descent into insanity:

Is this a dagger which I see before me,
The handle toward my hand? Come, let me clutch thee.
I have thee not, and yet I see thee still.
Art thou not, fatal vision, sensible
To feeling as to sight? or art thou but
A dagger of the mind, a false creation,
Proceeding from the heat-oppressed brain?

In this quote, Macbeth is tormented by the vision of an invisible dagger as a result of his guilty conscience. As he gets involved in killing more people who he thinks could imperil his position, his insanity only intensifies:

Prithee, see there! behold! look! lo!
how say you?
Why, what care I? If thou canst nod, speak too.
If charnel-houses and our graves must send
Those that we bury back, our monuments
Shall be the maws of kites.

In this scene, which is also called "the banquet scene," he sees the ghost of Banquo, whose assassination he ordered. Every guest is startled by Macbeth's reaction, and the scene implies that Macbeth's condition will only worsen.

As far as Lady Macbeth's mental decline is concerned, she becomes an irrelevant character once Macbeth gains power. Macbeth neglects his wife because he is obsessed with maintaining his status as the king of Scotland, so his wife no longer interests him. This reflects badly on her health, as we see in Act 5, when she is tortured by her guilt-ridden conscience:

Out, damned spot! out, I say!—One: two: why,
then, 'tis time to do't.—Hell is murky!—Fie, my
lord, fie! a soldier, and afeard? What need we
fear who knows it, when none can call our power to
account?—Yet who would have thought the old man
to have had so much blood in him.

She cannot forget about her involvement in the murder of Duncan, and her loss of inner peace proves to be too much for her to handle.

Good question.

To prove Macbeth's insanity, look at when he sees things no one else does like in Act 2 scene 1:

Is this a dagger which I see before me,
The handle toward my hand? Come let me clutch thee.
I have thee not, and yet I see thee still...

And later in Act 2 scene 1:

I see thee still,
And on thy blade and dudgeon gouts of blood,
Which was not so before.

And the ghosts…
Also, when he hears things no one else does:

Me thought I heard a voice cry “Sleep no more!
Macbeth doth Murder sleep”—the innocent sleep,

To prove Lady Macbeth's insanity, look at when she tries to rub the (invisible) blood from her hand in Act 5 scene 1.

To prove both, look at all the times they change emotions or positions quickly.