It is said repeatedly throughout Tolkien’s works in Middle Earth that hobbits are a hard and courageous lot in spite of their plump and peaceful ways. There is a core of hardness at the center of each hobbit, which with significant hardship will bloom into a courage and strength to be reckoned with.
But what is courage? Is it only a personality trait; a description of the nature of one’s actions? That seems a bit too simplistic, and I think “Fog on the Barrow-Downs” has something very powerful to say about the full meaning of having courage. Courage is not a simple word, a noun, an adjective about a person. It is not just a concept, a passive tag to identify what we do. In some ways it is a being unto itself, and ultimately boils down to one word: obligation.
When trapped by the Barrow-Wight, Frodo feels the intense draw of the Ring; it wants to escape and return to its master. It cannot be trapped and incorporated into the Wight’s horde. And so it puts tremendous pressure on Frodo to place the ring on his finger and escape. Now it can be argued that this is nothing to do with the ring and everything to do with Frodo’s fear. But I think his attempts to rationalize this potential action and his compulsive groping for the ring are evidence that the ring is very much active in Frodo’s thoughts and judgment.
One thing and one thing only pulls Frodo out of this temptation: courage. Courage is not about lack of fear or surmounting it. It is about obligation; knowing what is right and though the consequences and obstacles may be difficult and fearful, doing it anyways. And so, though disoriented and weak from his contact with the wight, Frodo strikes back, cuts off the wight’s hand and sings for Tom Bombadil.
Wandering Paths 
I’d say heading to Mount Doom and one’s own probable doom for the sake of defeating evil is rather courageous!
http://tinyurl.com/hobbitshirts
Yes it is!
But you have to remember the immense pressure that changed the hobbits and allowed their courage and heroic nature to bloom. They set out initially solely to reach Rivendel (which was in itself an act of courage, for a hobbit to leave the Shire). Almost all their adventure is forced upon them in some way, it is through this pressure that they flourish as heroes. A good analogy would be to think of how a diamond is formed. The same can also be seen in the Scouring of the Shire.
And I’d also take note, that Frodo’s decision, and Bilbo’s attempted acceptance of the task, to go to Mount Doom are both based on a sense of obligation: to do the right thing. So you could say, not only is this Hobbit courage we are seeing in this instance (above) but hobbit honor as well.