Rainbow Dash never leaves a friend hanging. When Soarin' gets cut from the Wonderbolts, the only thing he's ever had in his life, he falls into despair and self-denigration, fearful of his future. Rainbow is there to lift him up and formulate a plan to help him find a life beyond the Wonderbolts.
Perhaps more tellingly, Rainbow Dash will not save one friend (Soarin') by throwing another (Spitfire) under the bus. When she goes to see Spitfire, she doesn't start another argument or pin the blame. Rather, she takes Spitfire's verbal abuse and waits patiently for Spitfire's anger to tide over into frustration and anguish - then offers a shoulder for Spitfire to cry on. Rainbow understood that Spitfire had to cut Soarin', but necessity didn't make it any easier for her. Even though the office trailer was a gigantic mess with tossed papers, broken furniture, and shattered windows, and Spitfire continued to lash out at Rainbow Dash out of frustration, Rainbow Dash never raised her voice in response and kept the situation under control, allowing Spitfire to let out her emotions in a safe manner.
It even extends to the local jerk of the group, Shootin' Star. Sure, she admits that he's one of the bigger jerks she's met—herself included. Sure, he's been asking for that flank whooping for a long time. But he's still a teammate, and as such, when he's gone into full on depression mode after said flank kicking, she reminds Spitfire that he did earn the right to wear the 'Bolts blue and that they were a team and needed to support each other. Even him.
It also helps her eventually start more of Shootin' Star's Character Development, as while she's telling him the story about how she got Tank, she mentions that that's the reason that they kept being nice to Soarin' every time he screwed up. Because when you're on a team, you support them through the downs, as well as the ups. This actually makes Shootin' Star think.
Rarity, Twilight, and Rainbow Dash reconciling after the latter's harsh overreaction to the flight harness.
The Ponyville Pegasi's delight about the flight harness. Two stand outs include desiring parent-child bonding. One pony (heavily hinted to be Blossomforth) talks about how they want to show their parents how flower gardens look from the air and a second talks about how she wants to teach her Unicorn son how to fly just like her parents taught her.
Anytime we see any of the parents interacting with their kids.
Twilight and Rarity build their proposal to the Princesses around how the flight harness will help unite all ponies, especially mixed-breed families. At this point, the illusions they're using for the proposal include what appear to be models of Pinkie Pie and her daughters flying and playing together; Rarity tears up, and several reviewers comment on doing the same upon reading it.
The final scene of chapter 13 is one the most romantic and touching scenes in the story.
Similarly, the last scene of chapter 32.
Pinkie strapping on the flight harness in order to join the game her daughter Tootsie and Windfall are playing, being joined shortly after by Fluttershy. Then, Pinkie's infant daughter Puddin' joins in as well (apparently flying for the first time); Pinkie cries Tears of Joy at this.
When Twilight admits to being afraid of her and Spike drifting apart due to him learning to fly, he promises that he won't leave her behind.
"Dragons keep what they treasure."
At Soarin's suggestion, Applejack puts on the wing harness to go flying with him. As far as she's concerned it's a top-five contender for Most Fun I've Ever Had, and on top of it they finally tell each other "I love you" — well, AJ can't hear over the rush of the wind past her face, but she can read Soarin's lips well enough, and he has no trouble hearing her reply.
After Luna decides to sponser the Harmony Aeronautics team in the Pegathalon, they show their thanks by remodeling the flight harness and renaming it the "Luna Moth". Luna is moved to tears by this.
Soarin reassuring Spitfire in chapter 23 that he bears her no ill will for kicking him off the Wonderbolts and tells her she made the right call.
This troper forgets what chapter it's in, but there's one scene between Pound Cake and Twilight, which happens after the pegasi of Ponyville see the flight harness in action. He comes up to Twilight with a bag of bits in his mouth and asks her if he can buy one. As Twilight tries to explain that they don't have one in filly size yet he keeps talking, mentioning that he asked his dad if he could have an allowance and got a paper route so that he'll have enough money to be able to afford one harness if what he brought isn't enough. He eventually breaks down crying, saying that he never wants to see Pumpkin mocked for not being able to fly ever again. Twilight promises him that as soon as they make foal-sized harnesses, he'll be the first to know.
Pip has a Heroic BSoD after nearly being killed by the LaFish brothers, mostly over his self-perceived cowardice, which he views as failing to be brave the same way he's failed at all his other goals in life. Soarin' encourages him not to give up, and that no matter how bad things might seem to get, he can always climb back up. This not only pulls Pip out of the BSOD, but it also motivates him to get back together with Apple Bloom.
Soarin and Applejack agreeing to get married in chapter 31 manages to be this, a Funny Moment, and an Awesome Moment all at once.
Everypony's reactions to hearing about the above start as Funny Moments, before moving onto this once their initial shock wears off.
When Braeburn confesses his feelings to Rainbow Dash, she lets him down easy; she says she's flattered, but isn't at a point in her life where she feels she can settle down in one place, and also doesn't feel right making him choose between her and Appleloosa. That said, she does suggest that at some point in the future, she'll be ready and look him up. She then gives him a parting kiss.
Chapter 35 ends with a fluffy friendship moment between Champion and Charger. Something that Champion was in desperate need of.
Even though they're the villains (and, more to the point, real jerks), it's still nice to see that Flam (a.k.a. Professor Destiny) really care about his brother, when he looks for him in the sandstorm in Chapter 36.
It's also touching that Flam isn't bothering with the whole "Doctor Insanity" thing anymore since he's too worried about Flim.
Their reunion in the following chapter (though Flam goes back to their aliases afterwards).
Twilight finally accepting Spike's attraction to Quillina, and Spike assuring her that he's not leaving her behind because of this.
Turns out Silver Spoon hooked up with Snails, of all ponies. And the scene where he comforts her after she learns of Diamond Tiara's disappearance is quite sweet.
Soarin and Shootin' Star making amends, with the latter apologizing for being such a jerk to him, and acting as his second during his duel with Redtail.
Everyone wishing Soarin' luck in the duel, with all the mares giving him a good luck kiss.
Applejack's yell of sheer joy when she gets a letter from Soarin' saying he won the duel and is fine.
Applejack and Soarin's reunion at the end of the Pegathalon.
Soarin: Am I in Elysium?
Applejack: No darling. Yer in Equestria, with me.
Soarin: Even better.
Diamond Tiara tracking down Silver Spoon and begging forgiveness for all the wrong she's caused to Silver and all the other ponies she's known. You can feel the anguish and guilt as she apologizes, even saying that Silver has every right to still be mad at her. And then... Silver Spoon starts their old foal-hood Secret Handshake.
When Flim and Flam are brought before Celestia for punishment, they both individually try to take all the blame for their actions in order to spare each other. It's a small, but still touching moment, showing that for all their faults, they still care about each other.