In what ways are friendship and trust demonstrated throughout the novel To Kill a Mockingbird?

A great illustration of the themes of trust and friendship can be seen in the episode where Dill runs away from home. Dill's gone AWOL because he's tired of no longer being the center of his mother's universe now that she's remarried. So he runs off to Maycomb, where he seeks sanctuary at the Finch residence. Or, to be more precise, he seeks sanctuary underneath Scout's bed. Scout's pretty shocked to find him there, as one can imagine, but she's glad to see him all the same. Dill's a very good friend of hers, and she'll do whatever she can to help out.
Much to Scout's disgust, however, Jem goes and tells Atticus what's happened. This is a sign of Jem's growing maturity; he knows that Dill needs to go back home as soon as possible and that telling Atticus is the responsible, adult thing to do, but Scout doesn't see it like that. She thinks that Jem has broken the bond of trust that existed between himself and Dill and betrayed their friendship.

In To Kill a Mockingbird, friendship and trust are addressed in several ways, but the tenuous trust and friendship between Arthur “Boo” Radley and the children is the most interesting example.
An unlikely form of friendship in the novel is between the children and Boo Radley. At the beginning of the novel, the children cruelly dehumanize him by making assumptions about him, partly influenced by the intolerance in Maycomb. When Boo begins leaving the children small tokens, he wins them over and dispels their fear.

Boo was our neighbor. He gave us two soap dolls, a broken watch and chain, a pair of good-luck pennies, and our lives. But neighbors give in return. We never put back into the tree what we took out of it: we had given him nothing, and it made me sad.

The children realize that friendship is give and take and they had done nothing but take from Boo. Boo has been injured by a cruel father, and yet he still has a capacity for trust and friendship with the Finch children, despite their initial feelings toward him.
Atticus addresses the issue of trust in the novel when he says,

We're paying the highest tribute you can pay a man. We trust him to do right. It's that simple.

He is pointing out that society functions on the premise that we trust our neighbors, friends and family to do the right thing. Boo Radley trusted his father to care for him and was repaid with evil harm. Yet Boo functions as a trustworthy adult when coming to the children’s aid and saving their lives.

Friendship and trust are explored in the novel through various character's interactions with one another during particularly adverse situations. Miss Maudie displays her friendship and trust by supporting the Finch family before and after the controversial trial of Tom Robinson. She is quick to encourage Jem following the trial and stands up for Atticus' decision to defend Tom during the missionary circle. Jem and Scout's relationship with Dill also demonstrates friendship and trust. They not only play together but also get themselves into several precarious situations. The three children trust each other and stick up for one another whenever threatened with punishment. Jem is quick to deny the fact that they are acting out Boo's life story, and Dill lies to the adults after they raid the Radley's yard.
Other members of Maycomb's community portray the themes of friendship and trust throughout the novel. Sheriff Tate and some locals visit Atticus' home to warn him about the Old Sarum bunch, and Walter Cunningham's friendship with Atticus allows him to see the error in his actions. Boo also seeks to gain the children's friendship by giving them gifts in the knothole of his tree and comes to their defense when they are attacked by Bob Ewell. Harper Lee uses the character's friendship and trust towards one another to illustrate the small town, intimate setting of Maycomb.