The climax of the first book. Katniss and Peeta are the last two Tributes standing when the Game Makers sadistically revoke the rules change they made that would have allowed them to win together, meaning one of them has to kill the other. When Peeta tells Katniss to kill him, Katniss is so shocked and horrified that she tries to make him kill her instead. Her promise to Prim that she'd win, her promise to Rue that she'd win, her own desire to go home to her family- she forgets it all in an instant because she won't let Peeta die and go home without him. Even if she rationalises the stunt with the nightlock berries as her way of defying the Capitol's cruelty, even if she doesn't realise it yet, she already loves Peeta- she'd rather die with him than kill him and live on without him.
After Katniss smashes a bunch of plates in her room, the red-headed Avox shows up to try and clean them up. When she orders the Avox to ignore the mess, the Avox notices that Katniss has cut herself and cleans the wound for her.
Before Katniss enters the Hunger Games, Cinna spends a moment with her and tells her that if he could bet, he would bet on her.
Cinna: Good luck, girl on fire.
Katniss offers to share her sleeping bag with Rue. It's especially heartwarming when you consider how up to that point, she's treated almost everyone else like a potential enemy.
Katniss singing Rue a lullaby, then spreading flowers on her body, followed by the mockingjays taking up Rue's song, which, considering the symbolism of mockingjays, is also a Crowning Momentof Awesome.
Cinna's last gift to Katniss: a collection of outfits he's designed for her when she finally decided to be the Mockingjay.
Annie and Finnick's reunion.
No one seeing them could doubt their love.
On a similar note Finnick and Annie's wedding.
The people from the different districts doing their best to make the event special. Especially the District 12 people teaching everyone to dance and insisting on a special number of steps for the bridal couple.
On a related note to that quote, Katniss makes that comment and then promptly imagines her own reunion with Peeta being just like Finnick and Annie's.
In Mockingjay, Katniss giving Johanna the bag of pine needles. "Smells like home."
In Catching Fire, when Peeta mentions that nobody really needs him, Katniss finally realizes that she cannot live without him. At this point you know that she loves him for real.
"No one really needs me,” he says, and there’s no self-pity in his voice. It’s true his family doesn’t need him. They will mourn him, as will a handful of friends. But they will get on. Even Haymitch, with the help of a lot of white liquor, will get on. I realize only one person will be damaged beyond repair if Peeta dies. Me.
"I do," I say. "I need you."
Just a little moment, but I love when Primrose wakes up with Katniss in the middle of the night, cuddles under a blanket with her, and helps her figure out how she can keep Peeta and the other victors safe from the rebels at the beginning of Mockingjay. We've seen time and time again how much Katniss loves Prim, and it's sweet to see firsthand that it goes both ways.
The entire town giving its "respect to the dead" salute to Katniss after she volunteers in place of her sister.
And the repeat during the Victory Tour by District 11, (Rue's home).
"With my paint box at home, I can make every color imaginable..."
Peeta's showing of skill during the Quarter Quell, The painting of Rue.
Buttercup putting aside his hatred of Katniss in light of Prim's death, crying with her, and protecting her during the night so she can have what's implied to be the first good night's sleep she's had since the war ended.
As thanks for her treatment of Rue, District 11 sends Katniss a sponsorship gift of bread that was originally meant for Rue. Thanks to Peeta, Katniss can recognize the bread as having come from Rue's home and makes sure to thank them out loud so they know she understood and appreciated its significance. As far as Katniss can tell, this is unprecedented in the history of the Hunger Games: the first time any district has sent a gift to a tribute from another district.
Finnick in general. He's built up by Katniss as both an irredeemable manslut and a very dangerous enemy who won his Games when his sponsors sent him a weapon and he killed everyone in his path. As Katniss (and the reader) get to know him, however, he becomes charming, funny, tender, caring, heroic, pitiful, tragic, and sacrificing. He still has no qualms killing in self-defense, having gotten first blood in the Quarter Quell, but he plainly cares deeply for Annie, Mags, Katniss, Peeta, and Johanna.
Peeta saving Katniss as well as Prim and their mother from starvation in the flashback where he threw the burned bread to her instead of feeding it to the pig like his mother told him to. Katniss wonders if he'd burned the bread deliberately, even knowing his mother would hurt him for it, so he could give it to her; she dismisses it as impossible since he'd have no reason to do that for her, but we know better.
Countless moments between Katniss and Peeta, especially in "Catching Fire". Examples include the pearl scene, Katniss' despair when she thinks Peeta is dead, the moment they share at the beach, the day they spend at the rooftop together...
Peeta: I just want to spend every possible moment of the rest of my life with you.
Katniss: Come on then.
Every night Katniss and Peeta sleep in each other's arms. Whether you view it as romantic or platonic it's still pretty heartwarming.
Peeta running back to help Katniss after she dropped the tracker jackers. Not only does he risk getting himself stung again but he also blows his cover with the careers. Doubles as a Crowning Moment of Awesome, especially when he fights off Cato and takes a sword to the thigh because of it.
Katniss throwing herself into Haymitch's arms after getting out of the arena in the first book. The way their relationship has developed at this point is really touching.
"Nice job, sweetheart"
And before that, when she hears him yelling outside her hospital room:
...I can't help having a vague, comforting feeling that someone is looking out for me.
The can of lamb stew with dried plums in Mockingjay.
One word: "Always."
Before the rescue team leaves on the mission to get the victors away from the Capitol, Haymitch offers to have Katniss drugged so she can sleep through the whole thing. You wouldn't normally consider something like that a CMOH, but consider the context: As Katniss notes, Haymitch has been spending the last few decades of his life trying to find relief from the horrible memories of what the Capitol did to him and the only relief he's ever found is at the bottom of a liquor bottle. Offering to have her knocked out it is a gesture of mercy. Haymitch is trying to spare her from having any additional horrible memories of her own.
The end when Katniss and Peeta's children are playing. One of the chief fears of Catching Fire was that Katniss knew that not only would she be forced to marry Peeta, but their children would be flung into rigged Games. All that sacrifice, horror and pain wasn't for nothing. She married Peeta of her own free will, and their children, and the entire new generation, will grow up in a world without Games.
Neither the end of the oppressing regime, the end of the Hunger Games, nor the end of a life of just barely being able to get by was enough to make Katniss feel like she could start a family safely. What did, in the end? Peeta wanting children so badly. She loves him so much that she is willing to overcome her fears so that he can have the children he longs for. And she did it twice, despite having such emotional difficulty carrying her daughter. And also on the flip side - Peeta chooses a life with Katniss, even though for fifteen years that appeared to mean never getting to become a father. He loves her enough that he would rather have only her than have a family with somebody else.
"I'll allow it"
Even with the way it ended, it's nice in parts of Mockingjay where Katniss and Gale reconnected as friends, especially the little scene with the berries and their two-man "May the odds be ever in your favor!" exchange.
YMMV on how heartwarming it is, but it's pretty cute how Effie has become kind of a doting mother or aunt-type figure to Katniss and Peeta in Catching Fire. Especially seen when she tries to take them down to training, but Haymitch doesn't want them to look like they need a babysitter, so she just walks them to the elevator, fixes their hair, and pushes the button for them. It's possible that almost anyone who has a mother/guardian like that but isn't a little kid anymore can relate to the situation.
In Mockingjay, the fact that Katniss immediately decides to take up protecting her prep team after she finds them, captive and in terrible condition. "Because if there’s one thing I know, it’s that Cinna would never have approved the abuse of these three, who he managed with gentleness and patience."
In Mockingjay, we get a very sweet exchange when Posy asks about Octavia's green skin.
“It’s meant to be pretty,” whispers Octavia, and I can see the tears threatening to spill over her lashes.
Posy considers this and says matter-of-factly, “I think you’d be pretty in any color.”
The tiniest of smiles forms on Octavia’s lips. “Thank you.”
Katniss' prep team meeting her mother: they're actually quite nice to her and praise the way she styled her daughter's hair for her Reaping Day. Then, she proposes to show them how she did it and they happily accept.
The end of the movie, which implies that unlike his attitude in the book, Peeta wishes to remain friends with Katniss after the Games.
Thresh and Rue's relationship in the movie. The look on his face when he sees her on the ceiling... those who see the scene by itself can easily mistake them for siblings.
When it dawns on Katniss and Peeta that maybe, just maybe, they really will be able to go home again. Hell, any and all scenes in the cave count. "Then stay. Please?"
Every friendship moment between Katniss and Gale. They're a pretty awesome team.
Peeta running back to help Katniss after she dropped a tracker jacker nest on him.
Peeta making a joke that draws a smile from Katniss—one as genuine as the smiles Gale drew from her.
The developing friendship between Katniss and Rue, climaxing in the scene where they sleep beside each other, almost like real sisters. It's very short, but sweet as all heck.
'Oh Katniss... oh... you would've been the most beautiful bride.' Shows how much much more Katniss has come to mean to Effie - much more than just another tribute from the poorest district.
In the aftermath of the tragic loss of Philip Seymour Hoffman, which has utterly shaken the fandom, fans have organized specific times for three-finger salutes in memory of their beloved Plutarch Heavensbee.