gives the best definition of a mockingbird:

“Mockingbirds don’t do one thing but . . . sing their hearts out for us. That’s why it’s a sin to kill a mockingbird.” 

Tom Robinson is a mockingbird in that he doesn't do one thing wrong. All he does is provide pleasure to the folks he encounters. That is exactly how he got in trouble. Tom Robinson was helping Mayella with some chores. He was humming a melody as he chopped up the dresser drawers. He was giving of himself charitably. He was in no way harming anyone. 

Then when Bob Ewell came home and found Tom Robinson, he set out to ruin his life. Tom was innocently helping Mayella when Bob Ewell accused him of raping his daughter. The white community assumed Tom was guilty, thus killing Tom with their conviction before he even had a fair trial:

Tom’s goodness is recognized by very few of the characters in the book. His willingness to help Mayella with the many tasks she has to perform around the house is seen by the white community as inappropriate. They can not understand why a black man would feel pity or compassion towards a white woman. They themselves would not feel the urge to help an Ewell, the most despised of the white population and so, to them, the only reason for Tom to offer such help would be for a sinister purpose.

Boo Radley is considered a mockingbird because he had no evil intentions against anyone. Boo was misjudged. Jem and Scout judged him based on rumors. Boo is not evil. He helps Jem and Scout out many times. First, he left gifts in the knothole of the tree. He has a warm expression of his affection for Jem and Scout. He sewed Jem's pants when Jem got caught in the fence and had to come out of his pants. 

Boo Radley puts a blanket on Scout the night Miss Maudie's house burns down. Most importantly, Boo saves Jem's and Scout's lives. On the night Bob Ewell attacks Jem and Scout, Boo kills Bob Ewell in his effort to protect Jem and Scout from Ewell's evil attack. 

Tom Robinson and Boo Radley never had intentions to hurt anyone. They are caregivers. They provide beautiful music for those they try to help. They never did anything but sing their hearts out through their good deeds. They were protectors of others. Both Tom and Boo are misjudged based on evil rumors. It is a sin to to kill Tom Robinson but he dies.

Boo Radley cannot live a life of privacy due to the evil rumors that circulate about him. It is a sin to treat him with such disrespect. 

The idea of the “mockingbird” is used throughout the novel as a metaphor for people who either do no harm or who contribute positively to society. Their positive role is easy to overlook because we take for granted the kinds of things they do. While they may not seem original, in the sense that the mockingbird imitates other birds’ songs, neither do they cause any harm. During the course of the novel, Tom did not commit a crime, but Arthur Radley did, yet Tom was incarcerated and killed, while Arthur continued to live peacefully at home. The novel’s theme of racial injustice is carried forward in these two contrasting examples.

When Atticus Finch tells his children that it’s wrong to kill a mockingbird, the statement is foreshadowing of Tom Robinson’s death. This “sin” is a collective wrong that the people of Maycomb, through the prosecution and the jury trial, inflict on Tom, which ultimately results in his death. Tom had tried to be kind to Mayella, but the white people resented the notion that a black person felt sorry for any white person. In addition, many Maycomb people seemed to believe that black men were sexual predators and rushed to judgment about Tom’s actions.

Arthur “Boo” Radley apparently suffered from agoraphobia, as he did not leave his house. The Finch children and their friend Dill made up stories about him and tried to spy on him, but he responded kindly with overtures of friendship. When Scout and Jem were attacked, Arthur left his home to confront Bob Ewell, killing him in the process. Sheriff Tate declines to arrest Arthur, although he is sure that he was the killer. Tate anticipates that the prosecutor would not pursue a trial, and if it came to that, that Arthur would be exonerated so Tate does not insert Arthur into the legal system.

In Harper Lee’s To Kill a Mockingbird, a mockingbird is a metaphor for purity and innocence. Boo Radley, Tom Robinson, and Atticus Finch are all metaphorical mockingbirds in different ways.

Atticus tells his children that killing mockingbirds is sinful because they are harmless creatures that do nothing “but make music for [people] to enjoy.” Mockingbirds symbolize goodness and innocence in the novel, and their qualities parallel those of three of the story’s most important characters.

Despite the rumors circulating about him, Boo is a kind, caring, innocent being. He does not bother anyone, and he tries to be helpful when he can. For example, he does not hesitate to risk his own life to protect Scout when Bob Ewell tries to kill her. He does not want credit or recognition for his good deeds; he simply wants to be kind and live in peace.

Tom is another example of a metaphorical mockingbird. He is a good man who wants nothing more than to earn an honest living and take care of his wife and children. Contrary to Mayella Ewell’s accusations, Tom is a gentle, non-violent, pure soul. Unfortunately, the people of Maycomb are so blinded by hatred and racism, they wrongfully convict Tom of a crime he did not commit. Tom is fatally shot while trying to escape from the prison he should not have been in. The Maycomb townsfolk metaphorically kill a mockingbird when they convict Tom.

Atticus is another example of the personification of goodness and innocence. He is fair-minded, kind, and tries to help those in need (even at his own expense). He decides to defend Tom knowing that Tom does not stand a chance at a fair trial because of the color of his skin. Atticus knows he will receive backlash for his decision, but he does not let that deter him and he remains determined to do what is right in spite of the threats and abuse he and his family endure.

The prejudiced, racist, hateful views and behaviors of the citizens of Maycomb are illuminated and emphasized through their terrible treatment of the mockingbirds of the town: Boo, Tom, and Atticus.

Before fully considering the question, I think it's important to consider Miss Maudie's quote about mockingbirds: "Mockingbirds don't do one thing but make music for us to enjoy...but sing their hearts out for us." Thus, mockingbirds try to create beauty in a world that often doesn't appreciate them.

Tom Robinson encountered trouble because he tried to help Mayella Ewell. He saw Mayella's need, with so many younger siblings running around, and every time he passed by the Ewell yard, Mayella seemed to have some chore that she needed help with: "choppin' kindlin', totin’ water for her..." Tom swears that he never set foot on the Ewell property without a direct invitation from Mayella. When he stopped by on the fateful day that would forever change the course of his life, he did so out of an ongoing sense of kindness, because, as he puts it, she "looked like she didn’t have nobody to help her." Tom must have known that he was putting himself in harm's way to come to the aid of a white woman—alone and without witnesses—in this societal context. Yet his sense of honor would not allow him to refuse to help a woman in need. Tom's "music" was his compassion and kindness, and he extended this to Mayella, who was instrumental in ending his life. Tom tried to create beauty in Mayella's world by showing her the kindness that she seemingly received from no one else, yet ultimately this kindness was taken advantage of, and Tom was left to pay greatly for his "crime" of compassion.

Boo shows similar compassion toward the Finch children at the end of the story. His life is devoid of kindness; he spends it locked away in his house like a virtual prisoner. Yet when Scout and Jem need him, he breaks free to rescue them. He flies to their rescue in spite of the way society views him; he has been ostracized and has been the subject of town rumors for decades. Yet his "music" of kindness leads him to the Finch children, and he rescues them from certain death at the hands of Bob Ewell.

A mockingbird's purpose is to create beauty in its world; both Tom and Boo try to reach out to those around them in kindness and compassion to cultivate more beauty in the lives they have the ability to influence.

It's also a sin to kill a mockingbird. Thus, it was a sin for the prison guards to kill Tom, and it would be a sin to subject Boo Radley to the justice system for the same reasons.

I agree with the previous editor. The mockingbird 'sings'. Boo and Tom sing.

That singing is an expression of their innocence. So for Tom, he sings when he sits on the witness stand, takes the ridicule of Mr. Gilmer, and tells the truth when Atticus calls upon him to do so. It would have been easy for Tom to confess to something he didn't do and hope for a gentler sentence. Oftentimes a court acts compassionately on a criminal they believe has remorse for their crime.

Boo likewise sings. His song is not audible. His gestures intended to develop relationship with children communicate his innocence. He communicates his innocence by acting when he is needed. He saves the children from Bob Ewell and that is the biggest statement any man could make on behalf of kids.

The mockingbird symbolism is rampant in To Kill a Mockingbird. Tom and Boo are certainly two of the primary adult examples. I'm not sure I would use the "fly" terminology extensively, but Tom certainly yearns for the freedom to flee from his incarceration as you have mentioned. Both Boo and Tom have the characteristics of the novel's explanation of the bird: Mockingbirds serve no purpose but to make music and make others happy. They do not destroy crops nor prey on other creatures. They are small and innocent beings. Tom and Boo both fit this criteria. I suppose Boo "flies" when he leaves the safety of his self-imposed exile within his home to save Jem and Scout, just as he does on the other occasions when he ventures into the outside world to mend Jem's pants and cover Scout from the cold.

Like the mockingbirds Atticus tells Jem not to shoot, both Boo and Tom are gentle and harmless. They do not hurt or destroy. In their own ways, both men bring goodness and joy into the lives of others. Also like the mockingbird, both men are defenseless against forces that would destroy them.

Boo Radley has been victimized since his youth by his family and his community. Effectively imprisoned in his own home, Boo has been made the monster in Maycomb. Despite these injustices, however, Boo has remained a good man. He takes pleasure in watching the children at play, gives them gifts that fascinate and delight them, and grows very protective of them. He saves Jem's and Scout's lives, at the risk of his own. Boo's heroism is kept quiet so that he will not have to endure public scrutiny. Scout, wise beyond her years. realizes that exposing Boo would be like killing a mockingbird.

Tom is a husband and father who takes care of his family and feels genuine compassion for Mayella Ewell. Tom understands Mayella's loneliness and miserable life, and he treats her with kindness. In return, he is arrested for a crime he did not commit, tried, convicted, imprisoned, and then shot to death. Atticus had told his children it was a sin to kill a mockingbird; Tom's destruction is sinful in every way.