While there are lots of pages devoted to the idea that Hamlet's act of madness becomes true madness, I don't think you can make a case that Hamlet is insane. If insane means out of touch with reality, his soliloquys and monologues throughout the play prove him to be thoughtful and philosophical. He shows extreme emotion in his response to his mother's hasty remarriage and the task of revenge given to him by the ghost, but his assessment of himself and the situation are clear and logical.
In Act 1 he has a very perceptive understanding fromwhere the faults of men derive; in Act 2 he correctly evaluates his lack of action; in Act 3 he accurately deduces that people don't always take action because "conscience makes cowards of us all;" in Act 4 he admires the ambition and drive of Fortinbras; and in Act 5 he comes to a new understanding of life, stating that he realizes that there is a fate at work in all things, and that "the readiness is all." Only a sane man can work his mind through the this maze of thought and come to the fencing dual knowing that this could be the end of him, but he ready to face the threats and the fate that has brought him here. His final words, setting his election lights on Fortinbras, are logical as he seems to be trying to make things as right as he can for a Denmark that has lost its real leaders in this last act.
Hamlet is never a fool who makes no sense. He is never acting out of character unless he intends to. He consistently sees the truth and comments on it. These all seem the actions of a sane man.
I'm not sure those are the only two choices we have for Hamlet. He is probably not insane, though others think he may be so. We have too much evidence throughout the story--even right at the end of the play--that he's sane. He tells both Rosencrantz and Guildenstern and his mother he's only acting. The only major incident after that which might make a case for insanity is his grief at Ophelia's death. His final words belie insanity, as well.
As far as wisdom goes, Hamlet does probably do a few wise things--he does not rush to vengeance, nor does he trust his former friends Rosencrantz and Guildenstern, for example. The question is whether he does them out of wisdom or indecision or lack of resolve or some other characteristic. Wisdom, to me, implies a reasoned, well thought out response; instead, he seems to have one plan--act insane--but many falterings along the way. He doubts and questions and wavers all along the way. Hamlet doesn't seem particularly wise to me.
Perhaps he is both.