Hamlet is sane:

• When his father's ghost says that Claudius murdered him, Hamlet doesn't go ballistic and attack Claudius right then and there.  He thinks, [hey, this ghost could be lying to me.  Better make sure]
• To make sure that Claudius is, indeed, the murderer that the ghost claimed, Hamlet came up with the "play within the play".  No insane man could come up with such a clever plan.
• I like the theory that Hamlet played like he was insane, in order to fool those around him.  That way, no one would suspect that he suspected (and later knew for sure), that Claudius killed the king.  Just look at how many times he throws everyone off the trail.  One moment, he's sad, the next angry, then happy; he really keeps people guessing about him.
• He finds out that his so called "friends" are going to betray him, so he switches letters and escapes his intended demise.  Once again, do you think someone that had gone over the edge could do that?
• It is my own personal theory that Hamlet tried to hint to Ophelia that he was not insane.  Okay, so he did treat her a little rough, but if he didn't, people might figure out he was faking his unstableness.  Still, I think he tried to tell her that he was not insane; she just didn't catch on the his scheme.  Since she didn't figure things out, she went bonkers.  Oh well, Shakespeare had to bump her off somehow...
• Anyway, where was I?  Oh yes.  When Hamlet's "friends" (Rosencrantz and Guildenstern) first show up in the play, Hamlet is immediately suspicious of them, seeing that it might be a little too convenient that they just happened to drop by to cheer him up, and that maybe they're really working for the king. Since he is suspicious of them, he is cautious about how he acts and what he says.
• After Hamlet finds out that Claudius really did murder his father, he doesn't openly attack Claudius. Hamlet knows that if he is to avenge his father, then he must come up with a plan to kill Claudius without anyone knowing about it. If Hamlet were to openly attack Claudius, then Hamlet would be accused of treason, then executed. Hamlet is not real wild about that idea, so he waits for the right opportunity.

• Hamlet's distrust in his friends (Rosencrantz and Guildenstern) is totally uncalled for. Okay, sure, they really were working for Claudius. That's beside the point. The point is that two of his best friends come to see Hamlet and cheer him up, and he treats them like dirt. Hamlet is obviously paranoid. And when Ophelia dies and he finds out about it, he gets upset [because he still "loves" her], even though he is the reason that she went insane.
• Just look at how Hamlets treats his girlfriend (Ophelia). One minute, he says he loves her, the next that he hates her, then back again. He's either very indecisive, or probably just bonkers; no sane man would treat someone he loves like that.
• Hamlet is very unstable throughout the play. His mood swings are very drastic, sudden, and seemingly random; sane men don't change their attitude that the drop of a hat. He's not faking it [insanity], as the last section states; he is really insane.
• There are some instances in the play in which Hamlet almost gives himself away, one of which is the "play within the play". The "play within the play" is a double bladed sword. If Claudius ever figured out that Hamlet was the one that wrote the script, then Claudius would be very suspicious that Hamlet knew something he shouldn't. It's just dumb luck that Claudius didn't figure it out; all he had to do was inquire who came up with the play's plot. (personally, if I had just murdered someone and then saw everything I did acted out in a play, I would be very suspicious of someone; but Claudius wasn't, and I'm getting side tracked. Back to Hamlet)
• Hamlet's "To be or not to be" soliloquy can be interpreted to mean that Hamlet is contemplating suicide. If that's the case, then this is another display of his instability. If he were to kill himself, then his father would go unavenged and Claudius would be well off indeed. If Hamlet doesn't care about whether he lives or dies, he might as well kill Claudius, in the open where everyone can see him. Then, he could commit suicide or be executed for treason. Either way, it's better than killing himself and leaving Claudius alive.
• Okay, last reason and then I'm going to change subjects. If Hamlet was sane, he could have come up with a plan to avenge his father's death without endangering his own life; he probably could have done so fairly quickly. But, he doesn't. He waits, and waits, and waits, and by the end of the play, still no plan. Once again, it's just dumb luck that Hamlet is able to kill Claudius off before he himself dies.

Hamlet is not a wimp/coward:

• Hamlet is very with it when he talks to the ghost. A coward or wimp would have panicked. As it is, Hamlet had to run after the ghost before he could get it to speak; that was very brave, indeed.
• Hamlet pretends to be unstable, maybe even insane. He disregards what everyone thinks of him, just so that nobody finds out that he knows about Claudius. It takes a lot of courage to do that; could you honestly see a coward acting like that? No, a coward would have tried to hide away from everyone, or more likely run away from this dilemma.
• Hamlet plots and plans how to avenge his father for a large amount of the play. If he were really a coward, he wouldn't have even thought about risking his own life to avenge the late king.
• If Hamlet was a coward, he would not have had the nerve to treat his "friends" like he did. Hamlet was very quick to confront the dilemma of his old buddies working for the king.
• If Hamlet was a coward, he would not have been so quick to write the script for the "play within the play". He would think that the play might make Claudius suspicious. As it is, Hamlet did write the script, and was very successful in this endeavor.

• After the ghost told Hamlet that Claudius murdered him, Hamlet should have gone and accused Claudius, or even attacked him, but he did not, because he was afraid of the consequences. He does come close in a couple of scenes, and is very angry at Claudius, but backs off each time.
• Hamlet tells himself that he does not want to attack Claudius because he is not sure if the ghost was telling him the truth.  After Hamlet discovers that the ghost was telling the truth, he still does not make any headway towards avenging his father's death.
• He has an easy opening to kill Claudius while he is praying, but instead of attacking, he shies away.
• When Hamlet's "friends" stop by to cheer him up, he is instantly suspicious of them for no solid reason.  He does not have a clue that they are working for the king.  Hamlet is obviously paranoid.
• When Hamlet is confronting his mother, he notices someone hiding behind the curtains and stabs him with his sword, thinking it is Claudius.  This is a very cowardly thing, as Polonius (the man behind the curtains), could not see the attack coming.  Hamlet dealt a coward's blow.

• One theory is that Hamlet was a coward, and therefore very hesitant to act.
• Another theory is that Hamlet was at first waiting to make sure the ghost was telling the truth.  Later, he was waiting for the proper moment to strike.
• One idea is that Hamlet was insane; he just couldn't cope with the situation, so he had no idea how to deal with it.  Therefore, he spends most of the play trying to figure out how to deal with what's going on and what to do.
• No idea who first said this, but this idea is very to the point.  "No delay, no play."
• I doubt most literature teachers would consider this one valid, but the only person who could officially proclaim it wrong is Shakespeare, and he's dead.  Hamlet is lazy.  That's right, he is the biggest procrastinator you ever saw.  The real reason he doesn't attack Claudius is that then he would be king, and he would have to run the country.  As it is, he gets to goof off with his friends and have fun.