Booth's speech to Brennan in "The Parts in the Sum of the Whole".
Booth: Hear me out. Alright? You know when you talk to older couples who have been n love for thirty or forty or fifty years, it's always the guy who says I knew. I knew. Right from the beginning. I'm that guy, Bones. I'm that guy. I know.
Followed with a one-two punch with a Tear Jerker moment when she turns him down.
Hodgins, who has spent all of "The Bikini in the Soup" trying to find Angela a Valentines Day gift, finally comes up with one. He creates a hybrid slime that smells like roses and names it after Angela.
For that matter, the whole Valentine's montage, which includes the aforementioned, Cam finally going on her date with Paul, Clark getting over his doubts, and Booth and Brennan's 'Valentine's Day Massacre'.
Booth: Take that you dirty rat!
"King of The Loony Bin."
After discovering that he and Angela both carry a gene that gives their child a 25% chance of being born blind, Hodgins decides to take up piano and suggests Angela do the same with sculpting so they will be able to share their passions with the child if he/she can't see.
In "The Beaver in The Otter," Booth attributes the typical stupid frat behavior of the people involved in the case as basically, "you have to be bad and do stupid things to become a good person." Brennan begins to worry about this, since she's never done a bad thing in her life. Booth encourages her to run out on their bar tab. She does, and he's right behind, but not before slipping some cash on the bar. Brennan got her "bad" moment, the bartender didn't get cheated, everyone was happy.
Brennan reading to Russ's daughter in the hospital at the end of "The Knight in the Grid."
When Brennan decides to leave at the end of Season 5:
Bones: I can give you a list of forensic anthropologists who can do this job.
Camille: No, Dr. Brennan, you can give me a list of forensic anthropologists.
While saying their last goodbyes to Mr. Nigel-Murray, the Squints sing his favorite song as they load him into the hearse.
And even before that, Brennan bringing a potted flower to put on the casket.
Brennan: Did I do something wrong?
Angela: No, sweetie, you did something just right.
And the fact that it was Brennan, of all people, who knew his favorite song.
Try watching "The Babe in the Bar" after "The Hole in the Heart", and the short scene of Bones complimenting Vincent and praising him for his good work just makes her last words to him all the more convincing. (and heartbreaking, but that's a whole different Trope.)
The team gathering around Micheal Staccato Vincent Hodgins at the end of "The Change in the Game". It's the true essence of friendship. Even the kid's name is a CMOH! Michael Staccato Vincent Hodgins.
From the same episode, Brennan telling Booth that she's pregnant and he's the father. He just gets this grin on his face and looks so damn happy. Doubly heartwarming on a meta level. A lot of writers would have gone with a cheap shot—made Booth look ambigously shocked or stunned, and leave us with that to chew on over the summer. But that would have been unfair and OOC, and we had the entire summer to look forward to the happiness that we know will ensue.
After being told that Sweets had inprinted on Booth and Bones as parental figures.
Bones: What? Like a baby duck?
Booth: ...Come on Bones, we're going duck hunting.
Early on, when her then-boyfriend Sully leaves for the Caribbean and Bones is waving him off heartbroken, the camera pans out to show Booth standing behind her. Then he visibly cheers her up as they walk away.
During the show's first season, Brennan tended to be abrasive and empathy-challenged. However, at the end of "A Boy in The Tree", only the show's third episode, she was able to say just the right thing to the grieving mother of the boy whose murder Brennan and Booth had just solved:
"Ambassador Olivos, you told me that all a mother wants is to know that she's raised her child well, that your biggest regret is that you'll never know if Nester would've grown up to be a good man. But he was a good man. He died because he was trying to do the right thing."
When the team manages to re-unite the kidnapped deaf and mute abused child with her original birth parents in "The Signs in the Silence".
When Bones is stabbed by the Harbinger doctor in the season five premiere:
Booth: Bones! Bones, you alright?
Bones: Yeah, I'm fine. I'm okay.
Booth: Bones, your arm.
Bones: Oh God... [pulls the scalpel out]
Booth: Bones... easy, I got you. I got you, it's okay. It's alright.
Bones: He tried to kill me.
Booth: Alright, easy. Ambulance is on its way, okay? [holds her tightly]
Bones: Gotta keep the pressure on the wound...
Booth: I got it. I got it, just relax. Just trust me, alright? I'll take care of you. Ssshhh, I got you. Breathe. I'll take care of you. [kisses the top of her head] I'm right here, I'm not going anywhere, I got you. I got you, baby.
This promo for Bones Season 7, emphasizing the Booth/Brennan relationship, has Brennan changing her long-held stance on the concept of love(from the Season 5 ep, "The Dentist In the Ditch"), in what may be the most romantic example of Spock Speak ever:
"When Booth and I first met, I never believed that such a thing as love existied. I maintained that it was simply brain chemistry. But perhaps Booth is correct. Perhaps love comes first and then creates the reaction. I have no tangible proof, but I'm willing to accept Booth's premise."
Season 7 premiere where Angela shows Brennan a video of her and Hodgins when Brennan felt unsure about her blooming babyship.
Also in the season 7 premiere, the opening and closing scenes of Bones and Booth as a couple. As well as when they tell each other they love each other- Bones especially, and Booth's reaction ("I know").
Season 7 episode 2, where Booth and Brennan see the ultrasound video of their daughter.
from the same episode: what does a pregnant genius forensic anthropologist listen to while studying bones? She plugs her earbuds into a fetal heart monitor!Geek-Dawww!!
Also, when Brennan and the latest Squint Finn Abernathy bonded while analyzing a skeleton. Abernathy, a Southern-Fried Genius thought Brennan was teasing her when she spoke with a drawl, but she then told him, "I love John Wayne movies."
"The Prince in the Plastic": Booth reassuring Bones that she'll be a great mom, and Bones planning to spend quality time with her daughter...dissecting frogs. Also Bones and Booth playing with toys she bought together at the end of the episode.
"The Male in the Mail": Brennan is there to comfort Booth at the end of the episode when he deals with the death of his estranged father. David Boreanaz turned in a strong performance as he displayed Booth's grief.
"The Crack in the Code": When Booth and Brennan are examining their new house, and Brennan talked about planting a cherry blossom tree outside their unborn daughter's window. In general, the whole scene is heartwarming.
The team surprising Booth and Brennan in their new home when they return from the birth of their daughter.
Following in the tradition of Hodges and Angela's son's name, Booth and Brennan named their daughter Christine Angela Brennan, after Brennan's late mother and her best friend.
Parker meeting his baby sister in "The Warrior in the Wuss": initially, Brennan is concerned that Parker may be jealous of Christine, especially when she sees evidence of him cutting up family photos. At the end of the episode, Parker reveals that he was cutting up photos to create a mobile to hang over Christine's crib.
The fourth season episode "The Science in the Physicist" has a nice scene between Brennan and Hodgins, where Hodgins expresses his distaste for the Collar Institute, which Brennan once wanted to join but they rejected her because her specialty involves studying the past. This exchange in particular shows the genuine respect that the Jeffersonian team have for each other;
Hodgins: To eternity, to glory, to the future. (scoffs) Brennan: You disapprove of the Collar Institute? Hodgins: Up and forward are only two directions. Science should look in all directions. You taught me that. Brennan: I did? Hodgins: Every day. Brennan: Thank you.
The third season Christmas episode has a running sub-plot of trying to put together a family Christmas for the incarcerated father and brother of Bones. At the end, it's been set up so the brother's girlfriend and her daughters will visit them in prison and the brother will be allowed to wear civilian clothes (he had lied to his girlfriend's daughters about being in prison and told them that he was out of town on business). But no tree (risk of shivs). At the end, Booth shows up outside the fence lighting up a tree with his car's battery.
Bones figures out how to amuse an infant.
Bones: What would entertain you? (beat) Phalanges! [waving her fingers](beat) Phalanges phalanges phalanges! Dancing phalanges!
This gets a super cute callback when Bones does this with her own daughter, Christine.
At the end of season five at the airport as Bones is heading to the Maluku islands for an anthropological dig and Booth is headed to Afghanistan to train counterinsurgent army guys:
Hodgins: 'k, I made you this chart of all the poisonous reptiles and insects, what they look like, how to avoid them, and what to do if you get stung or bitten, so...
Bones: Thank you, Dr. Hodgins. I love you, too.
Bones: Booth informed me that the proffering of overly solicitous advice is indicative of love.
Bones: Don't be a hero, Booth. Don't be you.
All of the team's interactions with the victim's skull in "The Ghost in the Machine", particularly Booth saying a battlefield prayer over it.
In "The Maiden in the Mushrooms", Hodgins accidentally eats Abernathy's hot sauce - hot sauce that his now-dead grandmother had made. Feeling bad, Hodgins spends half the episode trying to re-create the recipe - and succeeds. The smile on Abernathy's face as he tastes the hot sauce Hodgins made says it all.
Booth placing his sharpshooter medal on the victim's casket at the end of "The Superhero in the Alley".
After discovering that the victim in "The Woman in the Garden" was a devout Christian, Bones declares that they are going to identify who the victim is and make sure that she gets a casket and a Christian burial.
Season 9, Episode 1. As if the writers were outright apologizing for Pelant railroading things last season, they mend the fences between Booth and Bones almost completely: Bones finally decides to put logic aside and have a little faith in Booth's reasons for calling off the engagement. Sure, she tells him he'll have to propose next time, but it's the thought that counts, as Bones is almost allergic to taking anything on faith and this is clearly a big leap for her. And he does.
Bones was given a chance to express her faith in Booth in "The Sense in the Sacrifice". In the face of overwhelming evidence that Agent Flynn, a man who once saved Booth's life, was secretly working for Pelant, Booth refuses to believe his fellow agent and friend had turned traitor. Bones immediately declares that Pelant had framed Flynn, insisting that her faith in Booth's instincts is rational; she compares her faith in his understanding of people to his faith in her skills as an anthropologist. Booth's reaction is to immediately kiss Bones in the middle of the FBI bullpen, thoroughly enough to reduce her to Buffy Speak:
In "The Blood in the Stones", documentary man Andrew Jursic shows interest in Caroline, which is reciprocated, and invites her out on a dinner. It is nice to see that love here isn't restricted to the younger generation.
The entirety of "The Woman in White," which focuses around Bones and Booth's wedding. Particularly adorable are their self written vows - including the Tear Jerker of Brennan's being the letter she wrote to Booth when she and Hodgens were buried alive - and then her adorable basically "Was that ok? Did I do that wrong?" question to Booth and Angela trying to set up a venue and get the wedding to actually happen the day it was planned, because their venue had burnt down the night before. It's a long story.
While Big in the Philippines has a few Tear Jerker moments, the way Brennan and Booth show their support to Wendell, who had just be diagnosed with Ewing's Sarcoma, a rare bone cancer makes you feel warm inside. To top off, after Wendell gives a Crowning Moment of Awesome when he says he is going to fight, Brennan gives him a massive hug, much to the happiness of Booth and to the shock of Wendell. Another one is when Brennan (who is one of four people to know) says that Wendell's work was invaluable, to which Angela responds that a broken arm is nothing. It speaks so much about how they all value Wendell as an anthropologist, not just another intern.
The end of the subplot of "The Heiress In The Hill", which deals with Hodgins finding out he has a mentally ill older brother whom his parents kept a secret from him living in an institution, where the funds needed to support him are drying up. Hodgins promises his brother that he won't be taken from his home, they'll figure something out, and they bond over their shared love of Twenty Thousand Leagues Under The Sea, walking off with their arms over each other's shoulders.
The show also manages a meta-moment of heartwarming as well. Almost every media portrays mental health institutions as horrible places, at best an undesirable place to be and at worst barely humane. Seeing one portrayed not only concerned about patients, but as an actual home for them is a refreshing change.
A veiled moment for Booth in the "Boy in the Time Capsule". Booth has to narrate a humiliating moment of his life to Bones. Long story short, he tells her about a boy in his high school who was... strange. Teen!Booth's friends hanged that boy upside down from a ladder and everyone laughed at him, including teen!Booth. Bones is confused as to why this is humiliating until he explains: that day, he could have stepped in and stopped all than. Instead, he stood and laughed. Consider: the moment of his life he is most ashamed is not him beating someone up or cheating on a girlfriend: it's him letting someone suffer. Also, he remembers the names of every girl he slept with in school. He was definitely notthat guy.
In "The Nail in the Coffin", Booth stopping the Sympathetic Murderer's suicide and promising him mercy from the law.
A weirdly sweet moment in the rising climax in the episode, "Prisoner in the Pipe". None of the inmates try to hurt a pregnant Bones while she goes to find the evidence needed from the real killer. They even guide her to the killer! Bones was right when she told Booth they wouldn't hurt because she was pregnant.
In season 9, there is a plaque of Vincent Nigel Murray on a pillar in the Jeffersonian. It's visible in a blink-and-miss moment of episode 18.