The first case she appears in, 2-1, where it's obvious from the get-go that Gumshoe fancies her and is really upset about testifying against her.
THE ENTIRETY OF CASE 3-3. Gumshoe ends up fighting tooth-and-nail to get her aquitted, and it's the only case (aside from 1-4) where he's completely, 100% on Phoenix's side. The discussions where he actually admits that he likes her, combined with his repeated attempts to make her lunch, just cements this ship. Also, the fact that, when trying to describe her, the best thing he comes up with is "She was a good cop" is heartwarming in a Fridge Brilliance type way: Gumshoe is so dedicated to his job, calling someone a "good cop" is probably the highest compliment he can think of to give a person.
A brief moment in 3-5: Gumshoe mentions that Maggey wants to buy him a new coat. D'awwww.
Case I-1, where again he's doing his best to stand up for her and comfort her when she's accused of murder (yet again).
Likewise, every time that Won The Case! plays. That music helps do an amazing job of wrapping up the story with a really satisfying happy ending.
From the manga, the end of one case where the defendant and her real mother are reunited after Phoenix proves both of them innocent.