Uncle Titus disarms Aunt Mathilda's anger over the things he brought to the yard (a pipe organ, a giant iron statue of a buck) by saying he intends to put the statue next to the house rather than sell it...as he puts his arm around her and says, "Now I have two dears."
Aunt Mathilda and Uncle Titus calculating and re-calculating the costs for Carlos's reward so that he can get extra money back. The second is particularly notable since it takes place after Carlos is forced to give the extra money to the woman his uncle sold Blackbeard to...so he pretends Aunt Mathilda made a mistake and gives him the same amount he lost.
The fact Shelby can't bring himself to harm the dogs his whistle keeps summoning. And Jupiter calls him on it, revealing he isn't the criminal he pretends to be.
Why does Jim Hall get rid of George's cage? Not just because he didn't want Eastland or his actors and crew to think the lion was dangerous, but because he wanted George to feel like a member of the family and felt awful locking him up, away and separate. Also, the scene where he and Jim wrestle and play together, ending with the lion purring. (Forget this is inaccurate, it's still adorable.)
Rory not wanting Angus Gunn's treasure to be found...because he wants to marry Mrs. Gunn, and is afraid she won't accept his proposal if she's rich (because he's just a poor Scotsman). When Hitchcock asks what the lady said about the matter, Bob says with a grin that she's thinking about it.
It's also a bit of a Tear Jerker, but Jupiter's simple, sad explanation to Hector Sebastian that Alfred Hitchcock has died in-story and that the boys miss him very much.
Constance Carmel's relationship with Fluke.
Both Peggy and Gordon Harker's relationships with Jupiter as adults, particularly Peggy noting at the end that after all the times she had to rescue Baby Fatso in the TV show, now it was finally his turn to rescue her.
Throughout the series, the relationship the boys develop with Worthington is made of heartwarming, to the point he considers himself the fourth investigator and always trusts them, giving them the benefit of the doubt in all situations. This is especially shown in how he reacts to Jupiter's kidnapping in Deadly Double and how he stands up for the boys against Mr. Temple and the police in Smashing Glass.
Also, the way their relationship with Alfred Hitchcock develops. At the start he's nothing but dismissive or antagonistic to them, and they in fact have to bargain to even get him to introduce their cases at all. But as time goes on he starts sending them cases or recommending prospective clients to them without even asking, genuinely praises their work, and is alternately amused, heartwarmed, or enthusiastic about various denouements to their cases. He outright apologizes in one of his introductions for his initial ill judgment of them (especially Jupiter, even as he admits he can still be irritating), and it becomes very clear that he has become very fond of them and enjoys their association as well. It makes his eventual death even more heartbreaking—and as much as Hector Sebastian clearly likes and admires the boys, and vice versa, something seemed to very much be lost after Hitchcock was gone from the series, perhaps explaining why it eventually petered out in the U.S.
No matter how many times the boys may fight or have disagreements, they always end up friends again—and are equally determined to solve mysteries so as to right wrongs and help those who need it.