What are some examples of Bilbo's leadership in The Hobbit?
During the first half of The Hobbit, Bilbo is just along for the ride. It is not until Gandalf deliberately leaves the party to go through Mirkwood alone, that Bilbo is forced to take leadership.
It begins when, after wandering from the path in the dark, the company become separated (Chapter Eight). Bilbo, all alone in the dark, wakes up to find himself being wrapped up by a huge spider. No one is at hand to rescue him. He fights and kills it himself, with his sword Sting.
Somehow the killing of the giant spider, all alone by himself in the dark without the help of the wizard or the dwarves [sic] or of anyone else, made a great difference to Mr. Baggins.
The next morning, Bilbo goes looking for the dwarfs and finds that they have been captured by a group of giant spiders and are all wrapped up, hanging upside down, waiting to be devoured. He embarks on a long and harrowing rescue, using his sword and his magic ring, that ends up taking all day. In the course of this rescue, he has to lead the dwarfs. They are dazed and frightened, and also sleepy and weak from the spiders' poison. Bilbo has to explain to them that "I am going to disappear" (for he had not yet told them about his ring), and get them to run away whenever he draws off the spiders. They also have to make several stands whenever the spiders catch up to them, and they must turn around and fight. This is the beginning of Bilbo's leadership.
That evening, when they are finally safe from the spiders but have not found the path again, they realize that Thorin (the lead dwarf) is missing. The next day, while wandering around trying to find the path, they are captured by a company of Wood-elves (Chapter Nine). Bilbo adroitly slips on his ring, making himself invisible, and quietly follows the Wood-elves as they bring their prisoners back to their underground realm. He sneaks in the gate, and thus begins his second episode of leadership. Slowly, over many days, remaining invisible the entire time, Bilbo makes contact with all the dwarfs in their cells and discovers that Thorin is also being held in an isolated dungeon. Then he conceives and carries out the famous escape by barrel. To pull this off he has to convince all the dwarfs to come out of their cells when he sets them free, and to agree to get into the barrels. They have to do this quietly, and with limited time before they are caught. Bilbo succeeds in all of this.
When the dwarfs arrive by barrel in Laketown, Bilbo gets them out of the barrels, listens to their complaints, and revives them. Then, since Thorin is now with them, leadership immediately reverts to him, and Mr. Baggins once again fades into the background.
The company make it to the Lonely Mountain and manage to find the secret entrance. Now the dwarfs once again treat Bilbo like a hired hand. Per their agreement, he goes alone into the tunnel, steals a golden cup, and brings it back to the dwarfs (Chapter Twelve). Then he tells them that although he has no idea how they are to get the huge mound of treasure out of the mountain, or in fact how they might get rid of the dragon, he is willing to make a second trip down the tunnel to see if he can find Smaug's "weak spot."
Naturally the dwarves [sic] accepted the offer eagerly. Already they had come to respect little Bilbo. Now he had become the real leader in their adventure. He had begun to have ideas and plans of his own.
After his second trip down the tunnel (during which Bilbo actually talks with Smaug), Bilbo returns to the dwarfs and they sit outside the entrance to the tunnel for a long time, talking. Bilbo feels a deep foreboding and suggests they go inside the tunnel. He has to urge them, but at last they agree to it. Once inside the tunnel, he still feels fear. He urges them to close the door. They are reluctant, but he finally gets them to close it -- just before Smaug comes flying around the mountain and blasts with his fiery breath the mountainside where they had been sitting. Bilbo has saved the dwarfs again.
After Smaug is killed (destroying Laketown in the process) and the dwarfs have taken over the Lonely Mountain, a standoff develops between Thorin, who has barricaded himself and the other dwarfs in the mountain, and the Lakemen, who are requesting that he give them a share of the treasure to make up for the loss of their homes. Now it is time for Bilbo's final act of leadership. He sneaks out of the Mountain by night and secretly offers a bargaining chip to the Lakemen. It is the Arkenstone, the most valuable object in the hoard and the thing that Thorin most desires. By doing this, he hopes to prevent a siege or battle, which is what seems to be developing.
Bilbo's offer of the Arkenstone is not leadership in the typical sense, as no one is following him. But he is taking independent, strategic action, putting into play an alternative solution other than the one offered by Thorin, the ostensible leader of the company. And Gandalf (who now shows up again) seems to approve: "Well done! Mr. Baggins!"