The showdown outside the jail on the night before Tom Robinson's trial, to me, demonstrates Scout's character fairly fully. Her impetuousness, her valor, and her overall humanity are put on display as she rushes into the circle of men and defuses a dangerous situation with honesty, directness, and accidental wit.
I never failed to be moved by Scout's compassion for others as evidenced by the final scene with Boo when she stands on HIS porch for the first time and does what her father always told her to do: stand in someone else's shoes. As she stands there and "recalls" recent events as Boo would have seen them you see a very smart, perceptive girl with a generous and warm heart.
Scout is clearly the child of Atticus Finch despite her pugnacious tendencies as a little girl. That she writes her memories of childhood and sorts them out is indicative of the highly rational child that she truly is during the narrative. In the early chapters, her attempts to reason with Miss Caroline Fisher about the Cunninghams and the Ewells stand out as they indicate her innate proclivity to logic.
Indeed, this scene sets the stage for Scout's dominant rationality. For, she possesses a logic that emerges throughout the novel as Scout asks her father about entitlement and later during the mob scene she applies its priniciples to Mr. Cunningham, she wonders why her father is nice to Mrs. Dubose, she does not understand why she is punished for bashing her cousin after he has insulted her father's honor, she observes the trial of Tom Robinson and deduces much on her own and from recalling her father's instructions on witnesses. Finally, it is Scout's great logical reasoning that puts together all her childhood lessons as she stands looking from a different perspective on Boo Radley's porch in the conclusion:
Atticus was right. One time he said you never really know a man until you stand in his shoes and walk around in them. Just standing on the Radley porch was enough.
For me, it's when she fights Walter Cunningham on the first day of school. It shows us how feisty she is and how much she wants people to do (what she thinks is) the right thing. I think this really typifies who Scout is.
I think the real Scout is revealed best during the Christmas holiday meal at Finch's Landing with her Uncle Jimmy and Aunt Alexandra. Scout has mixed emotions about their yearly visit: She loves Alexandra's food and enjoys talking with her Uncle Jack, but she hates her cousin Francis. Francis antagonizes Scout, calling Atticus a "nigger-lover," until she finally decides to use her fists, "splitting my knuckle to the bone" on Francis' front teeth. Her Uncle Jack spanks her, Scout cries and becomes angry, and the two later make up, with Jack apologizing to her for not listening to the whole story. Later, Scout eavesdrops on Atticus' and Jack's conversation; Atticus, knowing she is listening, eventually tells her to go to bed. Scout goes to sleep wondering how Atticus knew she was listening.