The first time Hikaru ever uses magic, she knocks down Alcyone, one of the most powerful mages in Cephiro. Complete with Theme Music Powerup in the anime.
Despite being grievously wounded, Umi wants to learn magic to save Hikaru from Alcyone. Not only is she successful, but Fuu learns magic to heal them both.
The battles in the Spring of Eterna, when the girls are attacked by those most precious to them and can't bring themselves to fight back. When they realize that they're only illusions (after a pep talk from Emeraude, who is rapidly losing her powers), they each invent powerful new spells and defeat them in one shot, obtaining the Escudo.
Fuu's battle with Caldina. Her friends have been hypnotized to kill her, so she invents a spell on the spot to bind them. Then when Caldina takes control of her, she breaks the enchantment through sheer willpower and invents a second spell to take her out. (In the manga, at least. The anime has Ascot helping out, so she only invents one spell.)
Hikaru vs. Lafarga, especially her Badass Boast in determining to win using only the sword. She manages to disarm him, despite the fact that he'd been curbstomping all three of them.
The girls defeat Zagato by combining their attacks.
The first time the girls summon their temporary weapons, assuming an Ass Kicking Pose. Even cooler when Umi realizes she's not the only one who knows how to fight and complements the others on their form.
After Ascot takes out Hikaru and Fuu with a monster immune to fire and wind magic, Umi is left to fight on her own and Ascot starts summoning even more monsters. Just as one of them is about to squash her, Fuu casts a shield spell and then faints. Even though she's so injured she can't stand, she invented a new spell just in time to save Umi.
Umi defeats about a dozen huge, dangerous beasts almost single-handedly... and then she marches up to Ascot and makes him apologize for insulting her friends. Then with an Armor-Piercing Slap and a Kirk Summation, she gets him to do a HeelFace Turn.
While the ending is tragic, the trio did manage to beat Zagato and Emeraude with barely any time to recover between the two fights.
During the battle between Eagle and Hikaru to determine which of them will become Cephiro's new Pillar, Eagle reveals his plan on what to do as the Pillar. He wants to relocate everyone on Cephiro to the surrounding planets and remain alone on Cephiro, which he'll turn into a dominant state, to prevent the tragedies of the Pillar system again. Hikaru immediately tells him how wrong he is and that his plan is the same as Emeraude's plan, which would hurt people. Her conviction and inner strength is so strong at this point, she becomes the new Pillar.
At the end of Rayearth II, Mokona reveals that it is God and declares that Eagle Vision will vanish for failing the Test. Hikaru challenges this. And Lantis threatens to kill it if it doesn't let them both go.
And then there's Hikaru dragging Eagle back with her, despite it being stated to be impossible. Then Umi and Fuu cross dimensions themselves and help her.
Fuu using her bow to break a rock that was summoning monsters when it was hundreds of feet away in episode 3, "Ferio, the Handsome, Mysterious Swordsman".
Fuu gets one in episode 7 of the anime when she slices the Monster of the Week in half as it tries to eat her, which makes her weapon evolve.
While the circumstances were tearjerking, Hikaru's weapon evolving in episode 8 as she was forced to kill a monster she'd befriended to save Umi and Fuu.
Umi's weapon evolves in episode 9, "The Magic Knights' Greatest Crisis" when she breaks through the Lotus-Eater Machine and accepts that she's got to become a Magic Knight. She makes it back just in time to save Hikaru and Fuu from Alcyone.
Selece's revival as a Rune God in episode episode 10, where he defeats the Monster of the Week by clasping his hands together.
Hikaru has one that lasts half of episode 14 wherein she chases a monster that abducts Umi and Fuu. She charges headlong through a forest of thorns, gets knocked into a canyon, falls three times while climbing out, and, despite being quite a bloody mess, still has the strength to take out the monster in one hit from her sword and a magic spell. Not only that—she figures out that the monster is able to smell out blood like a shark, so she cuts herself with her own sword in order to make it come back so she can kill it. Both Determinator and Plucky Girl don't even begin to describe it.
The reappearance of the Rune God's in episode 22, "Cephiro and the Three Countries". They explain to the girls, they too wish to protect the people of Cephiro and offer their power to Hikaru, Umi and Fuu. Also counts as a heartwarming moment.