Hamlet's Horatio is the only person in Elsinore who isn't caught up in the emotional turmoil surrounding Hamlet's supposed madness, as lampshaded by Hamlet himself: "Give me that man that is not passion's slave, and I will wear him in my heart's core, ay, in my heart of heart." As a result, he's the only one alive at the end.
In one parody of Hamlet, Horatio must relate the story of the play to two policemen who are checking to see if they need to press any homicide charges; Horatio himself states that he is now stuck with the exceptionally depressing job of arranging the funerals for and burying all of his friends and their families on his own.
It's from Top Ten Shakespeare Stories by Terry Deary, if anyone's interested.
Marcus Andronicus in Titus Andronicus, at least for a while - by the end even he seems to be losing it.
Leon Tolchinsky from Fools, the only character not to be infected with a curse of stupidity.
Benvolio in Romeo and Juliet. Other characters even seem to be aware of it. At two different points, an authority figure asks him to explain what happened, trusting he will be able to say so accurately.
Doc in West Side Story is a more dramatic version of this—he sees that the gang wars are ultimately meaningless, sees that Tony and Maria's relationship is going to end in disaster, and sees that absolutely nobody will listen to him.
Arsenic and Old Lace has an entire crazy family, the Brewsters. There are only two characters that know what's going on and are not insane, Mortimer and the unrelated Dr Einstein. (No, not that one.) And Dr. Einstein is a criminal accomplice in way over his head, and a little crazy too. So Mortimer ends up literally the Only Sane Man. He's ecstatic when he learns he's not related to the crazy people.
Peer Gynt has a moment in the fourth act of the play where the title character ends up in a madhouse, and ends up the Only Sane Man, given that even the keeper of the asylum is slipping.
1776 has John Adams often beliving himself to be this in his unceasing push for Congress to vote for independence. The problem is that he's obnoxious and disliked by near every other delegate regardless of which side of the vote they're on - Benjamin Franklin calls him out on it because Adams' attitude demeans the accomplishments of the other delegates.
In H.M.S. Pinafore the role falls to Dick Deadeye. Of course, since he's ugly, hunchbacked and named Dick Deadeye no-one listens to him, even when he's making common-sensical statements like "It's a queer world". In fact, at one point he manages to turn the rest of the crew's opinion right around by agreeing with them.