The theme song, as well as the visuals that go with it.
And it seems like all is dying and would leave the world to mourn In the distance hear the laughter of the last unicorn I'm alive; I'm alive
When the Unicorn decides to seek out others, just before she leaves, all of the animals of the forest gather at the edge and stare at her beseechingly. Remember the hunter's speech.
Molly Grue's speech when she first lays eyes on the unicorn, especially her incredibly gut-wrenching, "Where have you been? DAMN YOU, where have you been?!" and "Where were you when I was new? When I was one of those innocent young maidens you always come to? How dare you, how dare you come to me now, when I am this?!" Tammy Grimes' vocal performance absolutely has to be heard to be believed.
Schmendrick at first is annoyed Molly would berate a mystical creature like this, but the Unicorn understands completely. In an unspoken moment, she bows her head and Molly whispers, "That's all right. I forgive you." Then there's Molly's response to Schmendrick asking how she can see the Unicorn: "When you've been waiting to see a unicorn as long as I have..."
When Cully's band chases Robin and Marian and the Merry Men into the woods. Something about that ragged bunch of bullies and cowards believing so hard that they'd follow transparent mirages. Because we can't all be a hero or a unicorn.
Brigand: "Robin! Mr. Hood, sir! Little John! Will! Wait for me!"
But it doesn't take a hero to believe.
The sudden choke when Schmendrick rescues The Unicorn from The Red Bull by turning her into a mortal woman ... and being old enough to realize exactly what he had just done.
An odd one for those that are really empathic comes from King Haggard's speech, telling Amalthea just why he took the Unicorns. Though this might simply be another case of a magnificent combination of music and imagery. It truly says something about the quality of the writing, the music, the voice acting, and the animation that in this one scene Haggard is established as a despicable villain who is also a broken, pathetic, all-too-human man. Who, in their life, can not remember one moment of utter joy and bliss? And who would not do ANYTHING to make it happen again?
I like to watch them. They fill me with joy. The first time I felt it, I thought I was going to die. I said to the Red Bull, "I must have them, all of them, all there are! For nothing makes me happy, but their shining and their grace." So the Red Bull caught them. Each time I see the unicorns, my unicorns, it is like that morning in the woods and I am truly young, in spite of myself!
The end, where the heroes win the day and hundreds of unicorns, once thought to all but extinct outside the last one, appear out of the sea to reestablish their place in the world. The music for the scene is the finishing touch.
The main unicorn having to leave the Prince and once again become a unicorn, and the end line about her being the only unicorn who will ever know regret.
"I remember you. I remember."
Just the very thought that the unicorn would be alone having to endure human emotions is painful in itself.
Lir's death. Molly's reaction is heartbreaking but it's the unicorn's reaction that really digs deep and what forces her to put the Red Bull in his place. She was likely the first unicorn that truly had something worth fighting for.
Schmendrick. According to Beagle he's anywhere from being in his 40s or even his 60s, and he's a complete failure. Mabruck greets him "Schmendrick, my dear boy, how nice to see you", but when he finds out he's going to replace him, his open contempt shows through, calling him "The Runeless Wonder". It's plainly obvious Schmendrick feels deep pain every day of his life.
The unicorn's horrified reaction to becoming human
"I can feel this body dying all around me..."
Finding out that the unicorn had spoken to both Schmendrick and Molly, but Lir had only seen her, and Schmendrick and Molly wouldn't tell him what the unicorn had said.