The entire relationship Link and Zelda have in this game:
After winning the bird-riding contest, Link is flying back toward Skyloft when Zelda excitedly jumps out into the clouds. Link hurriedly swoops in to catch her, after which Zelda nonchalantly says that she's fine. It's a relatively minor scene, but it does a whole lot to flesh out the cute relationship that Link and Zelda have.
Back to that Zelda-jump scene: Someone considered it to be cut for the purposes of data space. Everyone else decided to cut something else instead. Even the dev team thinks it's adorable!
And it also shows how much trust they have. Because think about it: Zelda leaps off of Skyloft - an island high, high in the sky, a fall that would surely kill her if she didn't call for her Loftwing - and fully trusts Link to catch her! She's not scared or worried, she just jumps and knows Link won't let her fall! And when he does, she just says she's fine when she sees how worried he looks! THAT'S some serious trust they have!
Going with the trust thing, it's also apparent whenever Zelda pushes Link off of high platforms, he's not mad, she only does it when she knows he has a safe way of landing/not falling and when her doing so does have a negative consequence, she will apologize! Again, that is some serious trust!
Zelda is very adorable in this game! Heck, even Ghirahim called her adorable!
And Link's reaction to the news that Zelda is about to wake up. The poor guy has spent the entire game worrying about his best friend (and perhaps more), has seen her enter an thousands-of-years-long slumber (which did not go over well with him) - and now he finally, FINALLY gets to be with her again, knowing that she's okay. His expression just sells it.
Link and Zelda finally being together for the first time after she awakens from her thousand year sleep. ...It's subverted, however, when a total Mood Whiplash occurs and Ghirahim captures her.
The Ship Tease between these two in this game is just ridiculous. They're best friends, Link does whatever it takes to save Zelda, he does NOT appreciate Ghirahim harming or attempting to harm Zelda, his Death Glares are usually directed at him, they trust each other without question and would do anything for each other! And it also helps that in this game, they know each other. In games like Twilight Princess and Ocarina of Time, they didn't know each other enough to really be anything more than friends, acquaintances or allies. And in games like Wind Waker, Phantom Hourglass and Spirit Tracks, they're too young! But in this game, they're childhood friends and aren't too young! Make of THAT what you will!
In fact, Demise's final speech canonizes it when he says that all future Links will "share the blood of the goddess and the spirit of the hero". Babies Ever After it is!
Link's expressions in this game just make him so darn adorable. When Link listens to the various parts of the Song of the Hero, he looks so genuinely peaceful, which he probably hasn't felt much throughout the game.
Halfway through the game, Link is on his way to the Sealed Grounds when Groose hitches a ride on his Sailcloth. Groose starts freaking out when he sees the Chirris and Gorko, and he starts shaking Link and demanding answers. This is the guy who has been an absolute ass to Link for pretty much their whole lives, up to and including putting Link's life in danger by penning up his Loftwing. What does Link do? He gives Groose a pat on the arm, smiles, and proceeds to calmly explain what's been going on with him and Zelda and the Surface.
Even Groose's reaction to the offscreen Info Dump comes across as Heartwarming. He basically tells Link to head back home and get some rest, and that he can handle things from here. The Kikwi are all horrendous cowards, the Mogmas are self-centered treasure hunters, the Ancient Robots are worse than useless, and Impa gave him a "The Reason You Suck" Speech the first time they met. Groose is offering to help - indirectly, and entirely for his own reasons, but still. After the amount of hell Link has gone through, he knows someone else cares about and is going to help him save Zelda.
This game makes it so you'll do something special for at least one person no matter what you do. For example, Peatrice (the Item Check girl) gets absolutely no business at all... and then Link comes around. The more you go to the Item Check booth (and you will have to go there often to swap out or sell stuff), the happier Peatrice gets, and the more she starts crushing on Link.
This is especially nice for those who are sick of the Ship Tease between Link and Zelda in their many different incarnations, and this gives gamers a chance to give Link a female companion who is upfront with him.
Fi wishing Link good luck before the battle with Demise. Considering how mechanical and impersonal she typically acts, it's very touching to hear her finally encourage him in that way.
Fi's final words to Link before going to sleep forever and becoming one with the Master Sword is that she'll meet Link in a next life. She does. All Links share the same spirit, so every time another Link picks up the Master Sword, whether they know it or not, they are meeting again. So in a way, it really isn't goodbye forever when they part ways at the end.
There's a flashback moment in Breath of the Wild where Link is near death and Zelda is in despair - and then she hears a voice from the Master Sword telling her that Link can still be saved. Fi is still looking out for her master after all that time.
After the creditswe cut to a bunch of the villagers at the now earthbound Goddess Statue. Zelda says to Link that she wants to stay on the land the rest of her life, and Link doesn't even bother to answer with anything but a nod and a smile, implying he's going to stay with her. The way it's shot with the Triforce in the background and Zelda's white dress makes it look like an exchanging of vows.
Groose. When you leave him he's having a meltdown over being unable to do anything while Link takes on the Big Bad. The next time you come back he's not only gotten over his funk, but built an entire rail system to combat The Imprisoned. Talk about Character Development!
Going hand-in-hand with that is right at the end, when Groose travels back in time after Link to see what he can do to help, and managing to dive and catch Zelda's body after Demise tosses it aside. The simple fact that the player will think "Hell yeah, Groose!" about a guy who was the resident Jerk Ass at the start of the game speaks volumes about how far he'd come.
Pretty much the whole relationship between Groose and Link. In the beginning, Groose locks up Link's Loftwing, and basically just shows that they're rivals. After Groose follows Link to the surface, eventually, he warms up to him and thinks of him as a friend, helping him out against the Imprisoned.
And at the end of the game, when Link inserts the Master Sword into its pedestal, effectively locking Fi inside it, we see that Groose has come to stand next to Zelda and watches him do the deed. We never see his face, but him being included at all speaks lengths as to how their relationship has evolved by that point and how much Groose regards Link as a friend.
Similarly, Groose's relationship with "Granny", the elderly Impa. When he is down in the dumps about being useless following the first battle with The Imprisoned, she is the one to help him see that there is still worth in himself and that he has a role to play, even if it's not exactly what he thought it was. Groose's name for her, "Granny", is clearly spoken out of affection and he does whatever she needs done around the temple. He hugs her after reuniting with Zelda, and then gets in front of her protectively in order to defend her from Ghirahim. It's this, as well as his developing friendship with Link, that contributes the most to his development.
Compare the colour of Epona in other games to the colour of Link's Loftwing, apparently so rare it was thought to have vanished from the line. Perhaps Fi's not the only one who's been sticking around with Link.
From the Bazaar there's Bertie's idle animation. His kid he's carrying on his back starts crying, so he pulls a silly face and makes a bunch of goofy sounds until the baby is laughing happily. It's a very small but sweet moment.
There's also another small moment when you say to Gondo that you believe his grandfather's stories about the Ancient Robots. After ages of being teased about believing his grandfather the guy is thrilled when someone doesn't make fun of him and actually agrees with him.
The Love Letter sidequest can end in either Pipit and Karane getting together, which is adorable. Or it can end with the ghostly hand caressing a sleeping Cawlin. It's weirdly cute, but hilarious.
The Mogmas in general. When they hear about Link trying to save Zelda they'll give him useful items such as Digging Mitts or, in Ledd's case, his own Bomb Bag, to help Link on his journey and/or will give him encouragement and hope that he finds her alright. The Kikwi also do this to a slightly lesser degree.
When Zelda is revealed to be the goddess reincarnated, and has to enter a slumber to protect herself and to maintain the seal on The Imprisoned/Demise, she asks Link to wake her up when he has collected the Plot Coupons, as she always woke him up when they were kids. Most dialogue trees have a "no" option, but all the choices here are "Of course!" "I promise." and "I will." This also doubles as a full-on Tear Jerker.
When Link frees his Loftwing after it's been kidnapped by Groose. The way it spreads its wings and opens its mouth, as if to say "I'm free!" and the affectionate, gentle way it nuzzles Link are quite heartwarming, especially given their special connection and the fact that they grew up together.
When you first see the trapped bird, the poor thing is freaking out. The moment Link steps into its sight, it calms down. The very moment the bird sees its owner, it knows it's going to be okay.
An adorable callback to that scene: there's a Gratitude Crystal in that place at night.
There's a lovely harmony between all the shopkeepers in the Skyloft Bazaar, as none of them actually talk down to Linknote although some of them do give you some unpleasant looks if you inspect something and then choose not to buy it, and every shop is recommended to Link by the owner of another shop at least once. Purchasing anything but ammunition from Rupin has him recommend you have it upgraded at the Scrap Shop, trying to purchase one of them when your pouch is full has him advise you to drop something off with Peatrice, Luv advises you to boost her potions at Bertie's cauldron, Gondo praises Sparrot's fortune-telling after you get Scrapper patched up... After the Malo/Chudley/Gorons business competition in the Twilight Era, it's nice to see shopkeepers recommending one another to their customers.
The Gratitude Crystals in general make for a lot of these. Miniquests aside, there is one in Zelda's room, like special, personal thanks to you.
Groose's lackeys, Cawlin and Strich, don't seem to care that Groose goes missing from Skyloft and seemingly never returns. Cawlin mocks Groose behind his back and Strich is more interested in secretly hoarding insects. But then they come down to the Sealed Grounds with Gaepora after the credits to see Groose before leaving with him. Aww, they really are friends.
There's something very gratifying about Beedle thanking you for completing his side quest, and hearing his trademark thankful voice clip used with sincerity.
At the bird-riding contest, they announce that it is the 25th anniversary of the contest. Given that this was the 25th anniversary game, there was much squeezing.
Kukiel, the sweet little girl who befriends Batreaux, is utterly precious in her innocence. She even encourages Link to collect all the "grabitude crackles" for her "Uncle Bats."
The long ending cutscene before the credits roll has a subtle example shortly after Link, Zelda, and Groose return to the present. There's a shot of the Master Sword in its pedestal, right where Link left it in the past. The doors behind it then open on their own, reminding us that just beyond was the room where Zelda had been sleeping. So in a sense, the sword (and Fi, in a sense) were watching over her almost the entire time.