Arguably, any time Lightning and Serah talk about or to one another, and whenever old party members from XIII show up to help save the day.
Noel's speech about his motives and what he wants at the start of the game. He wants a future that has people in it, given how bleak and depressing his time is and how he was the last person born and the last person left after everyone died.
The ending starts this way, with Fang and Vanille safe, and the possibility of Serah finally being reunited with Snow and Lightning. Then, of course, everything goes to hell. But up until that point, everything was awesome!
Noel reveals to Caius that the reason Yeul keeps up the process of reincarnation is just to be with him again. To be clear, Caius isn't a Nice Guy, but damn if you don't kinda feel for him during his breakdown.
A sidequest allows you to find out more about Mog and where he came from. Apparently he'd been playing with his fellow Moogles until he got caught up in a time paradox and lost his memories in the Void Beyond until Lightning saved him. You eventually get to a time gate where he remembers everything that will allow him to return to his time and his own kind, but that means he'll be leaving Serah and Noel and their journey behind. He chooses to stay with them.
Mog: Mog has a new home - right by Serah's side, kupo!
Serah and Lightning's final conversation in Valhalla shortly after Serah's death. Not counting the mostly informational meeting in New Bodhum 700 AF, this is the first time they were able to talk personally in years, and Serah's speech to Lightning about how she's happy and doesn't care that she's dead is very emotional. Serah happily snuggling with Lightning before disappearing makes it all the more touching.
A sidequest in the Vile Peaks has you locate six soldiers crashed there because of a paradox. When you have located Blitz, the sixth and final soldier, he immediately recognizes Serah as Lightning's sister. When asked how he knows Lightning, he explains that he and his buddies were once Lightning's colleagues in the Bodhum Security Regiment. Lightning's sudden resignment and subsequent rebellion against the Sanctum ticked them off at first, but they eventually came to understand that she tried to save Cocoon and, inspired by her brave actions, they decided to apply for positions on the Academy's force. They named their unit the Blitz Squadron in her honor and some of the members even took on a moniker related to Lightning. To seal the deal, Blitz says that while she may be gone, he believes in his heart that he and Lightning are still on the same team, trying to make the world a better place... It's only a short conversation in a big game, but it's a touching one.
A minor one: return to A Dying World after freeing Noel from his dispairing dream world, and walk all the way back to the empty settlement. The entire area is completely barren and empty, but when you get there you find a herd of wild chocobos - not the paradox kind either. Normal, yellow Pulsian chocobos. Immediately after that Noel renews his conviction to save the future. It's minor and optional, but it's definitely a relief for the player, who at that point has just played through the bleakest level in FF history.
5 "The Future is Hope" is the happiest alternate ending and happiest ending period in the game. It ends with Snow and Serah riding off into the sunset on a brand new (yet related) adventure through time.
"Beneath a Timeless Sky" is more subdued. Sure, Serah and Noel are trapped in a desolate world but Yeul is with them and she doesn't have visions anymore so she's safe. The overall mood is one of hope: things are bleak now but they'll get better.'
The DLC Sazh's Story: Heads or Tails? starts off with Sazh flipping a coin, with the slow-mo title shot displaying the two faces: the goddess Etro (Heads) and a dark skull (Tails). Time freezes, he gets a bit of exposition from the casino owner, he catches the coin, and then flips it again because he didn't get a chance to catch it properly. Slam, peek... "What the...?" And it's a chocobo. Cue Chocobos of Cocoon in the background as Frocobo flies up to him. That falls under Funny.The Heartwarming kicks in when he amasses enough Fortune Medals to reunite with Dajh. That is beautiful enough, with them reenacting the Nautilus scene from XIII and Dajh NOT crystallizing, but then the casino owner tells them that "the possibilities have begun to disappear". Mystery Man calls a couple of coin showers onto the Chronobind table, and they're all coming up tails. The skull. "The possibility of the coins coming up heads has disappeared from the world. [...] The future has begun to disappear." Sazh then comes just shy of quoting Vanille: "Even if I my actions won't make a difference, I have to at least try." He grabs a coin and flips it, slamming it against his hand with Dajh and Frocobo watching... then he peeks, and a smile rises on his face as he reveals the coin to the owner and Dajh. It's another chocobo. If that doesn't scream Take a Third Option, I don't know what does.