Near the beginning of the "Duck, Duck, Goose", Carter proudly trots out a 111 IQ score, resulting in Zoe being ridiculed at school. Fast forward to the end of the episode, when she gets her own results back, we see her opening her own score paper and seeing that she recieved a 157. When Carter asks her what she got, she then responds saying a 112.
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.
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. This unfortunately leads to the Downer Ending when he gets absorbed by the Artifact and dies.
When Jack's ex-wife visits in "Unpredictable", 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.
"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.
While Jack is concerned when Joe and Henry disappear, he seems genuinely devastated when Fargo vanishes.
Stark's final moments in "I Do Over". He's ready to sacrifice himself, and while his hubris and bluster remains intact, it's clear knows there's a very good chance (possibly even a dead certainty) he won't survive. He's warm and complimentary to both Carter and Fargo, specifically asking Carter to be there for Alison, and to make sure she gets the logic diamond necklace (in which he recorded a holographic message for her, but that won't be revealed for another episode). Special note to his final words to Doug: "Keep pushing buttons, Fargo." On the surface, just an admonition to keep working so they can fix the problem, but going deeper, Fargo's known for "inappropriately pushing buttons" and causing problems. Stark's telling the little guy to keep being who he is and doing what he does. Stark may have talked down to Fargo a lot, but he did truly care for and respect him.
"Show Me the Mummy", 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.
"Welcome Back, Carter", Jack getting his job as sheriff back by the end of the episode. All thanks to Henry, and a little help from Sheriff Andy's encyclopedic knowledge of the town charter.
"Insane in the P-Brane", Lexi's boyfriend Duncan and Fargo becoming good buddies.
Taggart's return to Eureka in "Have an Ice Day".
Zoe takes her SCAT exam at Tesla, which is 99.8% accurate at determining where someone's best aptitudes lie. She's decided she wants to be a doctor, but the SCAT tells her she's best suited for robotics. Her boyfriend Lucas tries to be supportive by buying her a robotics kit, and while Zoe does eventually get it to work, she still states that medicine is her passion, and Lucas decides to stop pressuring her, telling her "I knew you weren't like 99.8% of people already."
"Founder's Day", Jack finally kisses Allison... in the year 1947.
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."
In O' Little Town, Allison's explanation to Carter why she goes so all-out every Christmas. Her parents were wonderful, and they had Christmas, but her parents were very practical scientists who didn't talk about Santa or wrap presents (waste of paper). So Allison wants to give her kids what she didn't have as a child. She wants to give them magic.
Zoe figuring out Jo, who's been bah-humbuging the whole episode is everyone's Secret Santa, giving them absolutely perfect gifts completely anonymously. When Zoe confronts her about it, the explanation is beautiful. Jo's favorite thing as a kid was to give perfect gifts to her brothers, and watch their faces light up. Now, they're all in the military and deployed, Jo's family hasn't spent Christmas together in years, but she carries on with her Eureka family.
In "The Ex-files", Illusion!Jessica Lansky congratulating Fargo for standing up to General Mansfield and overcoming his insecurities.
Lansky: (to Fargo) How do you feel?
Fargo: Considering I'm about to lose my job? Not to bad.
Lansky: (smiles knowingly just before vanishing) Then my work here is done.
In a very bizarre way, once Zane figures out that he and Jo have a history of which he is entirely unaware (thanks to time travel shenanigans), his relentless pursuit of her becomes this. He's not after her because she's hot (though she totally is), but because it honestly bothers him that he doesn't know what they were to each other, what they meant to each other, and the potential happiness they could be missing out on together. In his own snarky way, Zane really does love Jo and wants her to be happy.
The fact that Fargo is so protective of Zoe, even going so far as to confront Zane about her directly.
Jack admitting that it was actually kind of nice to see Stark again. Proving the two men did have an actual friendship underneath their rivalry.
"Omega Girls", Jo encouraging Zoe when she starts to worry she wont be able to save Jack. Zoe in turn encouraging Jo to stop Beverly and her thugs.
Jo: Atta girl. You can do this.
Zoe: Okay... Now go get that bitch.
After passing the written test in "Of Mites and Men", Jo breaking into a Happy Dance.
The little bit of Fargo's backstory we get to see in "This One Time at Space Camp...". It shows that Fargo 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.
From "One Small Step..." "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.
Despite the zaniness, "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 disappeared 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.
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.
Jack: When everybody found out what you were up too, they wanted to help out. Kevin: This is amazing...
Jack convincing Andy to help get the parts needed to build the machine.
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, Jack tries to apologize to Fargo 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.
The first thing Warren does upon his return to Eureka in "Jack of All Trades"? Give Jack and Allison a big hug. He even calls them his "favorite" couple.
After over four years, Jack finally proposes to Allison.
After the Roller coaster 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 Too Deep", after nearly drowning to death at the bottom of Lake Archimedes. Jack and Allison finally get married.
"Smarter Carter", 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."