Carter's note from "Noche de Sueños"... "Jo, glad you're back on your feet. C." Plus a pair of dancing shoes.
Jack and Zoe's reconciliation in "Right as Raynes" after she tries to run away from home.
Zoe: You said you were done with me.
Jack: Zoe, you're my daughter. I will never, ever be done with you.
The recreation of the Nobel prize ceremony for Dr. Thatcher in "Dr. Nobel". Years ago he was emotionally and mentally crushed by not receiving it, breaking it off with the woman he planned to marry and ending up a babbling old man in Eureka's nursing home. He gets up to the podium and looks out across a computer generated audience, every detail the same, and seems to be talking to himself again. Then his girlfriend waves from the crowd and he smiles, going into the speech that he's been holding in his pocket every day since he lost the award.
"Games People Play": Everyone in the town stops by to check on Carter.
Abby: What are you all doing here?
Vincent: We're here for Carter and Zoe
Abby: All of you?
Vincent: Well yeah, I mean, he's our sheriff and she's our girl.
When Jack's ex-wife visits, he refers to his time in Eureka as "The best year of [his] life." For all his complaining and exasperation, he loves his new town.
Jack finally kisses Allison... in the year 1947.
Poor clean freak Dr. Carl Carlson from "Invincible" makes friends with Jack, telling him his the first friend he's ever had. Then he saves Jack's life. Unfortunatly this leads to the Downer Ending when he gets absorbed by the Artifact and dies.
It's pretty heartwarming to see all the effort Fargo puts into the Nathan Stark Memorial Hall. It shows just how much he idolized his late boss.
"I'm sorry ...sometimes I forget you can't see me smile". An incredibly Daw worthy moment...provided by an A.I. and an android.
Carter: He wanted me to tell you that you're his Casita Bonita.
The little bit of Fargo's backstory we get to see in "This One Time at Space Camp...". It shows that Fargo's had the drive to never stop trying to be a successful scientist, all because of one lesson he learned in his youth. You only really fail, when you stop trying. Years later Fargo has one of the top jobs in Eureka and is actually a respected scientist in his own right.
Jack taking some time to sit down with Fargo at the end of "Force Quit" after everyone from the Astraeus except Holly is back safe and sound and tries to apologize for not telling him sooner.
When Fargo is trying to deal with Holly's death in "Friendly Fire", Parrish, who is normally Fargo's rival and a massive Jerkass, comes in and offers to play a game of Dungeons & Dragons with him.
In "The Story of O2," in addition to her other problems, Jo's house is destroyed by a malfunctioning rocket. She moves into an open jail cell, but Carter won't have it. He tells her to get her stuff so she can move into his house's spare room.
"Sarah would enjoy the company. I would enjoy the company."
Despite the zanyness, "Do You See What I See" had lots of genuine heartwarming moments.
Jo, initially irritated by her "Jo White" getup and cadre of singing birds, becomes genuinely upset when the latest shift in reality makes the birds disappear, wondering whether the most likely imaginary birds dissapeared or just flew south for the winter.
The talking sled dog, that gives Andy some sage advice on why it's good to be an individual and not always part of a pack/group.
Allison tells her brother Marcus that in some ways, Jack is the smartest man she knows. He replies "He married you, didn't he?" It's heartwarming enough on its own, but it becomes meta-heartwarming when you realize that Marcus is played by Salli Richardson-Whitfield's real-life husband.
Also from that episode, the B plot involves Carter suddenly becoming super-intelligent - because Kevin doses him with a smart drug while trying to help him pass muster with Marcus. Kevin really wants Carter to be his dad.
The series finale has them all over the place, but a few stand out.
Holly's description of Jack's purpose in the town:
Holly: He's the strong force. He holds it all together. Zoe: He always does.
Carter finally not snarking about how he feels about Eureka, while about to toss himself inside an unstable wormhole:
(to Allison) "This is our home. And this is what I do."
After the Rollercoaster that is Ex Machina, the very ending, full stop.Turns out, the memory lattice Henry designed to pull Holly from the GD mainframe worked before the super laser-pointer she was hiding in went critical, as we discover when he plugs it into SARAH, allowing Fargo to see her again.
In "The Real Thing" once word gets around that Kevin is building a machine to track the lost Astraeus crew, the entire town turns out to help.