Emily's behavior that could be considered scandalous begins with her unwillingness to allow her dead father to be buried. The town is shocked when she won't release the body to allow for a funeral. She refuses to accept her father's death.
When Miss Emily starts taking rides in her carriage with Homer Barron, the town is both disgusted and surprised. Homer Barron is a Yankee, Miss Emily's father was a staunch supporter of the Confederacy, and just because the South lost the war, the town feels that Emily has abandoned her heritage and it is a disgrace.
At the end of the story, when the town discovers the dead body in the bedroom and realizes the Miss Emily must have slept next to the rotting corpse, the town reacts with horror.
The narrator, who represents the voice of the townspeople brings up the fact that she refused to pay taxes; not only this, but she kicked the men out of her house that had come to collect them. This was completely unconventional, and worth a mention several times throughout the text.
After her father died, they find it scandalous that she would just keep one servant in the house, and a male at that. They gossiped, "Just as if a man--any man--could keep a kitchen properly." They think that a woman, especially a "proper" woman should keep more servants about.
Her behavior with Homer Barron scandalizes the entire town. She was upper-class, and he was working class, and heaven forbid they be seen together, especially without a chaperone. They start to gossip, and feel that she had become a "fallen woman." The narrator states of it that "Some of the ladies began to say it was a disgrace to the town and a bad example to the young people." They think that relatives should come and stay with her, to convince her of the error of her ways.
In the end, they pity "Poor Emily" in most that she does, and find her a perverse curiosity, but her unusual stubbornness and strange ways do scandalize the town quite often.