Wyldstyle: You have the ability to be the Special because I believe in you.
Emmet invented the double-decker couch so everyone could watch TV together and be buddies!
Uni-Kitty is a walking Moment Of Heartwarming while she's her cute little self. You can't help but go "awww" when she does/says something adorable.
She's still adorable when she goes into rage mode, as this happens when she notices Emmet is in danger.
Uni-Kitty: Emmet, GO! GO! Now's your chance!
In the video game, Metalbeard wonders why Bad Cop is helping them. Uni-Kitty responds by saying "he's nice now". Metalbeard accepts this.
"If you trust him Uni-Kitty, then that's good enough for me."
Bad Cop scribbles back on a good face in marker. His scribble-face is adorably happy.
Benny's belief in Emmet when it seems all hope is lost, with all the Master Builders and Emmet imprisoned.
Benny: Emmet, you'll... you'll think of something, right? Like you always do.
Added effect when said this later on:
Benny: Emmet had ideas...
Lucy finally opening up to Emmet and confiding in him that she wanted so much to be the Special. Emmet, having been gawking over her appearance throughout the movie, is now looking at her with respect.
Despite the fact it leads to a tearjerker a second later, Benny's continued faith in Emmet after they've been captured and Vitruvius just revealed that there is no such thing as the "Special".
Benny: Emmet, you'll think of something, like you always do...
Just before Emmet does his Heroic Sacrifice, he turns to Wyldstyle and tells her "It's your turn to be the hero." Knowing he was going to die, he decides to fulfill her dream before he goes.
foreshadowed by the psychedelic vision when Emmet touches the piece of resistance: the last thing you hear (very quietly) is Finn's voice saying, "now it's your turn to be the hero".
Emmet seeing everyone else being creative, and realizing that everyone is special.
The Man Upstairs.That moment when he doesn't have to be the bad guy.
The apology hug. BOTH of them.
Emmet: (gets hugged by Lord/President Business) Oh, we got a hugger.
Batman ultimately putting together who is this "Lucy" Emmet seemed to be referring to. This probably allows him to figure that he isn't so suitable of a boyfriend to Lucy.
Scribble-Faced!Good Cop getting to reunite with his parents, whom he never wanted to use the Kragle on in the first place. He lost his original face for them, and it was sweet seeing they don't care what he looks like and are just happy to be with him again.
Ma and Pa Cop: Son! You're OK!
They do genuinely care about him—they are absolutely horrified for their son when he gets his Good Face wiped off with nail polish remover.
The scene where Good Cop / Bad Cop is forced to use the Kragle on his parents is mostly horrifying and sad, but it's made very clear that Bad Cop does not want to do this, and he really does love his mom and dad.
Benny yelling with joy when he finally gets to build his awesome, slightly retro spaceship. All the time he was denied the chance, and now he has it. It warms the heart to see the little blue guy running around as he does the one thing he always wanted to do.
Wyldstyle convincing the citizens to stand up to Lord Business by telling them all about how Emmet, who was seemingly nobody, came closer than anyone to saving the day.
WyldStyle's reassuring the LEGO world of their potential within them and apologizing to them for looking down on ordinary LEGO citizens.
Note that all the LEGO folks that admitted to not really noticing Emmet at the the start of the film are all gathered in that spot watching the footage of Emmet in amazement.
Lucy becoming his girlfriend at the end. Bonus points for not believing that Emmet was worthy to be the Special, and then seeing his potential and starting to care about him.
When Emmet first meets Finn, he's terrified he is going to eat him. Instead of harming him, Finn smiles.
Finn: Hi Emmet.
Emmet: Uhhh... Hi?
Finn defending Emmet as his choice of the hero.
Earlier when Emmet first sees Finn playing with Lego's, he calls him adorable.
When Finn's dad actually takes the time to look at what his son built and asks him, "You built all this?". He's genuinely impressed.
Emmet is sitting on the double-decker couch he built with all his newfound friends. President Business is even sitting with them! Quite a clever contrast in the opening scene where Emmet is likewise sitting on his regular couch but with only a plant and his breakfast, and no one else, when obeying the Instructions' step of sitting "with your friends."
Adding to that, President Business had earlier critiqued the couch idea, saying that the worst place to sit would be the bottom. Guess where he sits at the end? He apparently volunteered to take the "worst" place, signifying his change.
The movie itself is a great Fridge Moment of Heartwarming. LEGO has existed in its modern form since its patent in 1958, well near sixty years ago. This movie does its best to bring together both the beauty of the original building kits and the wide assortment of licensed properties LEGO has made sets from. Anyone who has ever enjoyed LEGO can sit down and enjoy this movie. Few things have the power to bring together people like that. And it's awesome.
The best part: this movie was a massive success. People embraced it. Not to mention, the Lego toy sales went through the roof thanks to it's praise. And now they've became the #1 toy manufacturer and brand thanks to it's sucsess.
Another Fridge Heartwarming thing: Okay, so Finn picked Batman, a goth looking girl and a robot pirate to be among the heroes in the main group, right? But who else did he pick? An outdated minifig (Benny) and a homemade unicorn cat. And the hero? A generic worker. Not only was Finn happy to play with an old toy and a "girls' toy", he made a generic worker the hero! This kid doesn't discriminate!
You...don't have to be the bad guy. You are the most talented, most interesting, and most extraordinary person in the universe. And you are capable of amazing things. Because you are the Special. And so am I. And so is everyone. The prophecy is made up, but it's also true. It's about all of us. Right now, it's about you. And you... still... can change everything.
If Business is based off of Finn's dad, that speech might just be Finn's way of telling his dad all of this.
There's also the fact that rather than try and confront Emmett, after the speech, Lord Business takes it to heart, does a Heel–Face Turn, and willingly put the Piece of Resistance onto the Kragle himself.