What are some examples of Hamlet's bravery?

Hamlet displays his bravery when he goes onto the castle ramparts at night to meet with the ghost his friend, Horatio, and the castle guards have seen there. It could be that the ghost is the restless spirit of his father but it could also be that it is some evil spirit that will "tempt [him] toward the flood," as Horatio fears (1.4.77).
When the ghost beckons to him, Hamlet readily agrees to follow the spirit, saying, "And for my soul, what can it do to that, / Being a thing immortal as itself?" (1.4.74–75). He does not fear that the spirit can endanger his soul, the most important part of himself, because his soul is as immortal as the spirit is.
Despite Horatio's concern that the spirit could "draw [Hamlet] into madness," or otherwise endanger his life when he's been separated from his friends, Hamlet feels as though "each petty arture in [his] body [is] / As hardy as the Nemean lion's nerve" (1.4.82, 9–293).
The Nemean lion is a creature from Greek mythology that could not be wounded by any mortal weapons, so the lion must have been a very brave creature indeed, as he was thought to be invulnerable (except to Hercules), and Hamlet compares himself to that lion here. Not only this, but he actually threatens the men that try to hold him back, saying that he'll make ghosts of them if they do not "Unhand" him. This seems brave, indeed.

For a character whose indecisiveness is so commonly commented upon, Hamlet actually displays ample bravery throughout the course of the play. His decision to meet with his father’s ghost could be considered an act of bravery, as could his choice to duel Laertes—a well-known capable duelist—in the final scene. The biggest critic of Hamlet’s bravery is Hamlet himself, who laments his own inaction:

This is most brave,That I, the son of a dear father murdered ...Must, like a whore, unpack my heart with wordsand fall a-cursing ...

Hamlet openly questions himself and the audience: “Am I a coward?” (2.2). He even paints his own refusal to commit suicide as an act of cowardice: “Thus conscience does make cowards of us all” (3.1).
Per usual, Hamlet might be a little hard on himself. In a way, he is such an eloquent speaker that he has convinced scholars over the past four hundred years that he is a coward. But Hamlet’s bravery becomes clear when we look at what he is being asked to do: believe a ghost; commit regicide; take over the throne of a warring nation. Every step Hamlet takes in the play is an act of bravery, as oblique as they might seem. He is a quasi-suicidal college student who is stuck at home watching his mother marry his uncle. One could make the argument that for Hamlet, getting out of bed every morning is an act of bravery. Calling Hamlet a coward dismisses the difficulty of his circumstances. He may be logical, dramatic, and pious (demonstrated by his refusal to murder Claudius during his prayer), but he is a far cry from a coward.

Despite Hamlet's tragic flaw, which is his hesitation an inability to act, he demonstrates bravery at several crucial moments in the play.
One could consider Hamlet brave for approaching the enigmatic ghost that resembles his father. Despite the apparent danger, Hamlet follows his father's ghost and discovers that Claudius assassinated King Hamlet in the orchard.
One could also argue that Hamlet demonstrates bravery by casting his emotions for Ophelia aside in order to manipulate and fool Claudius into believing that he is an unstable madman. Hamlet understands that he is in great danger and demonstrates bravery by not allowing his love for Ophelia to intervene with his revenge plot.
Hamlet also acts bravely after discovering that he will be executed in England and cleverly writing a letter that leads to Rosencrantz and Guildenstern's deaths. Hamlet's choice to fence against the talented Laertes is also a brave decision—as well as his tragic final act of killing Claudius in order to avenge his father's death.

Hamlet is brave in ways that are more apparent to a modern audience than to Shakespeare's own. Yes, he openly pursues the Ghost despite warnings from Horatio and the soldier/guards, but his "bravery" is most interesting in his willingness to confront his own nature. He is a thinker who is being compelled to become a doer, and in the famous soliloquy "How all occasions do inform against me," as well as his reaction to the First Player's speech, he is urging himself to become what his situation demands that he become.
One interpretation (and I believe the correct one) of his behavior to Ophelia, often seen as cruel, is that it is his brave attempt to distance himself from her. He does truly love Ophelia, but she cannot share in what he plans to do, so she becomes yet another lifeline in his world that he must abandon.
Is Hamlet "brave" in the final scenes, when he agrees to fence with Laertes, then kills him as well as Claudius? That is less bravery than savage revenge. But he faces his own death bravely, without knowing what is to come. "The rest is silence."