Emily's secret in Faulkner's "A Rose for Emily" is that she apparently poisons Homer and keeps the body in her upstairs bed. What's worse, the hair the men find at the end of the story matches Emily's, which means that she has been sleeping next to the corpse.
Homer disappears from the town years before Emily dies and the corpse is discovered, but everyone simply thinks that Homer sneaked away because he didn't want to marry Emily.
Emily's secret is easily kept because the narrator can only reveal what the townspeople experience from the outside of the house. Since no one ever goes upstairs in the house, no one knows Emily's secret.
Emily's secret was found in her bedroom. She had presumably killed Homer Barron all those years ago and left him to rot in bed. But not only that, she was sleeping next to him every night since she killed him. That's a pretty major secret!
To see this, look at the last four paragraphs of the story (one is just one sentence). You can see that his corpse is there, kind of fused to the bed. And we know that she was sleeping next to him because there was a dent in the pillow and one of her hairs.