Dirk's treasure: a small chest filled with golden hearts.
Last Train to Blue Moon Canyon
The glowing yellow lights that surround the train are Nightmare Fuel—unless you read the in-game pamphlet that explains the original owner of the train saw them as a manifestation of his wife's ghost, come to say hello and celebrate the train she loved.
"Above all, my dear niece, let nothing happen to my train. It holds wonderful things." There's just something so adamant, so dedicated, in the way Lani Minella says it.
Nancy puts it best - Jake Hurley's greatest treasure wasn't gold, but his uncanny knack for making friends. He has a letter from his good buddy Abraham Lincoln, sent on the day he was assassinated, which turns out to be the treasure the villain is after.
Curse of Blackmoor Manor
The game is terrifying, and the titular clan seem cold and detached- until you get through the moving rooms, and reach the family's ancestral laboratory. There are messages from dozens of generations there, all recorded in the same book, all welcoming the newest heir and telling him/her You Are Not Alone. The room itself is...ancient, but not in a scary way. You can really feel how important family is to the Blackmoors, and how their traditions have bound them throughout the ages. It's a pity certain children of the family would never have gotten to see it, but it's still a very sweet moment that marks the game's shift to a more optimistic tone.
Mrs Drake's reaction to the fountain being turned on again. Turns out she's quite the Kuudere.
Alibi in Ashes
Alexei Markovic's conversation with Nancy after she gets out of jail.
Deirdre, end game conversation- best subversion of Freudian Excuse this troper has ever seen. Of course Deirdre hates Nancy and tries to annoy her, but that doesn't mean she wants to frame her. She knows that her rivalry with Nancy isn't worth committing a crime for, and the two actually go on to have a civil, honest discussion. Thank you, Deirdre, for showing more common sense than nearly all of the villains in the series. (It was also a neat way of outlining the 'shades of grey' theme- there's a difference between being a jackass and being a criminal, and both characters accept this.)
The Silent Spy
Nancy's flashbacks with her mother, especially with the sheet music. Kate wrote themes into her song representing her daughter, her husband, and herself. They're used to solve a puzzle, as a bonus.
Her growing relationship with Carson (her father) throughout the game. Carson starts off trying to get Nancy to come home, but after she and Ned confront him, they begin to open up, culminating in a long conversation over the end credits.
You can find Kate's final letter to Nancy in the villain's secret lair at the end of the game. Heart-rending, especially as Kate suspects she's going to die.
Nancy's final letter (the wrap-up at the end of the game) is written to Kate. If you find Kate's letter, you get a very sweet monologue from Nancy.