Monk/Heartwarming

In "Mr. Monk Takes Manhattan", when Stottlemeyer confronts a New York police captain for concealing information essential to investigating Trudy's murder; specifically, he said "my friend's wife!" This was the first time Stottlemeyer ever called Monk his "friend," and for long-time watchers is one of the biggest "awwww" moments.
The episodes "Happy Birthday, Mr. Monk" and "Mr. Monk and the Dog".
Both of these just got blown out of the water by the Series Finale. Not only does Monk uncover the secret behind Trudy's murder, not only does he survive, but Randy gets promoted to police chief of Summit, New Jersey and moves there to be with Sharona. We also see Monk getting much better about his neurosis. Oh right...and his wife's daughter is totally fine, becomes a huge part of his life, and manages to make him extremely happy and even continue his work, despite having no reason to anymore, because other people out there need his help. Oh, and just in case you were looking for a moment from the other two major cast members, Natalie has a crying fit of joy because she can see Monk getting better (though not so much that she'd lose her job), and Stottlemeyer even gets a brief scene where he kisses his wife before heading out to solve yet-another case with his awesome friend Adrian Monk. This is all aside from the brief flashback sequence we have of all the best moments from the series, set to some very heartwarming music as things wrap up. So yeah...a fairly large departure from the downer ending people were expecting.
"Mr. Monk and the Kid"
In the Season 2 episode "Mr. Monk Goes to Mexico", Stottlemeyer gets a call saying Monk has died. He declares that if Monk doesn't get a police officer's funeral, even though he was technically no longer on duty, he would quit.

Stottlemeyer: "I'm going to tell you something Randy, and I'm not ashamed to admit it. I loved that man."

- Of course, the moment ended up spoiled when it was revealed that Monk was actually alive, and the guy who died was actually a lookalike, to which Stottlemeyer ended up shouting that he hated Monk.
Also, during a Barbecue held by Stottlemeyer (which was also the first one Monk ever attended since Trudy's death) in Mr. Monk and Mrs. Monk, he mentions that he thinks that Monk is the man in a rather heartwarming speech while focusing on the grill. Unfortunately, that one actually backfired due to the fact that Monk was away trying to get supplies, causing him to not only literally be talking to air, but Randy Disher misinterpreting the comment as being directed towards him.
Stottlemeyer trying to convince Monk not to plant evidence.

"Don't do it."
"Don't do what?"
"Break my heart."

Pretty much any episode that ends with Monk fondly remembering Trudy with a happy smile. But one in particular always gets me right here, and it's not even a happy one. He has finally found Trudy's murderer. The man in question is on life support. Monk hits a switch, and as the man begins to gasp in agony, says, "This is me, turning off your morphine." Then he flicks it back on, and says, "and this is Trudy, turning it back on." Cue waterworks.
In "Mr. Monk Takes His Medicine," Monk is imagining Trudy as he longingly hugs her pillow to his face. He tells the imagined apparition that he can smell her lotion on the pillow, even after all these years.

Trudy: You never even liked that lotion.
Monk: (in tears) I love it now.

The episode that ended with him doing the fond remembrance of his therapist also does it. They had to write a new therapist into the show, as the actor playing Monk's therapist passed away and this was a really sweet way to honor his passing.
A smaller one, but one that not only warmed my heart but made me laugh: The end of Mr. Monk Goes to Group Therapy, when Harold Freaking Crenshaw says "Hey! My friend was talking!" and lets Adrian finish.
The entire scene with Molly on the beach in Mr. Monk and the End.
- Another mild one from "Mr. Monk and the End": when Stottlemeyer tells the entire police department that they won't be paid overtime for finding the guy who poisoned Monk, one of the officers replies "We're not doing this for the money."
  - Mild? That's a big moment, given that the police department developed over the series to merely being amused by Monk's eccentricities and annoyed when he got in their way to respecting and appreciating him enough to do that for him. Not to mention when the one cop says he's not in it for the money, everyone else nods their assent.
  - Relatedly, Stottlemeyer and Natalie's reactions to finding out that Monk could very well die- even though they get frustrated by him on a regular basis, it's clear just how much they both love the man.
Mr. Monk Meets His Dad. Monk and his father at the end with the bicycle.
- Additionally, his father stating to Monk how proud he is of his son just a few minutes before the ending.
A small one, when Stottlemeyer helps get Adrian out of a funk by calling in a favor and getting him a couple hours time in a clean-room lab setting. Adrian Monk, sitting in the cleanest surroundings one can get.
When Stottlemeyer gets married again, he asks Monk if he doesn't mind him marrying a woman named Trudy. Monk answers something like: "Of course not, everyone should have a Trudy in one's life".
The ending of Mr. Monk and the Magician. Monk and Natalie are watching Kevin Dorfman's video will (Kevin had been killed earlier in the episode), and it turns out that Kevin is rambling on in it in his characteristic way. Natalie asks him if she should fast-forward.

Monk: No. (smiles) Let it play.

Sharona tells her son he might be bullied at school soon because a bad guy owns some embarrassing photos of her. Little Benji is obviously shocked, yet he tells his mom she shouldn't let a bad guy win just because he might have problems at school.