A number of the character episodes, especially after the finale. Special mention goes to "Friends Forever," "The Family Ties," "Sorrow and Joy," and "Witch No More".
Arianna and Luke's first meeting as well as when they part ways.
The first time Misthallery fogs up, Layton, Emmy and Luke comment on it, with Luke suggesting that they hold hands so as not to get lost. This is heartwarming coming from poor little Luke, who up until meeting the Professor, had been so withdrawn and unhappy, as it shows him starting to open up and trust the Professor and Emmy.
From London Life:
When Layton sends your character on the final quest of the main storyline (which will allow you to win at LL), he gives a sweet little speech about how you're the perfect choice for the mission because it can only be achieved by someone truly happy - and you are the happiest person in all of Little London because your happiness comes from doing kind deeds that bring joy to others. Suddenly, all the time spent trying to catch specific fish or pick just the right flower seems totally worth it.