Ais regularly gets these during raid missions, such as piercing through an entire wave of monsters to give her teammates breathing room as well as single-handedly taking down a raid boss monster no one had encountered before.
Bell managed to defeat the Silverback a monster from the 11th floor, on his own. Every spectator watching cheered him for it.
Before we really knew Lyu's backstory, she manages to save Bell from a thuggish adventurer by giving such a threatening aura that the thug backed off.
Episode 8 of Season 1 has Bell facing down a minotaur, a creature he is mortally afraid of due to nearly being killed twice by them, in one of the most badass fights of the entire series. Every minute of it was brilliant and showcased an amazing fight befitting of a hero.
The best part comes right at the end. After breaking the Minotaur's BFS (which he was wielding one handed), he goes underneath the beast and jams his knife in its chest. He then casts Firebolt through the knife and into its body three times (snarling out the words), roasting the thing from the inside until it explodes (allowing him to reach Level 2). The members of the Loki Familia present (all of whom are at least level 5) can do nothing but stand there in awe.
Note that the above fact about the Loki familia is literal. They are described in the novel as being physically unable to take their eyes off the fight, as it perfectly symbolizes the life and death struggle that an adventure is supposed to be. They're watching the ideal of every adventurer in the city, and it stops them cold. And, in the next episode (and the ones following the equivalent in Sword Oratoria), this galvanizes the group that witnessed the feat into being even better fighters.
A relatively minor Awesome moment: Bell shatters the record for reaching Level 2 the fastest, by doing it in one month. The previous record was one year, and its holder was Aiz Wallenstein.
Episode 13 of Season 1 has a full-on raid being fought, where we see everyone doing their best to overcome a titan of an enemy that has absurd abilities. Everyone contributes in their own way, building to a climatic finish where an Bell uses a new Argonaut-empowered skill (Hero's Strike) to cut the thing in half.
The Siege against Apollo Familia in volume 6 of the Light Novels. Having to assault a castle defended by nearly a hundred adventurers with only five people. And yes, Hestia Familia wins.
Hell, Book 6 provides every member of Hestia Familia (temp and otherwise) with at least one Moment of Awesome!
Lili gets two: first, she becomes the first person to ever fight off the effects of Soma's wine. This act so impresses Soma that not only does he let Lili leave his familia, he drastically changes the way he treats and interacts with his children. Then, during the War Game itself, Lili fine-tunes 'Cinder Ella' to such an extent that she successfully impersonates a specific member of Apollo familia for several days, and manages to execute the equivalent of the entire Trojan Horse strategy - both as the Horse and the strike force inside it - by herself, thus pulling off one of the greatest spy games in Orario's history.
Welf talks Hephaestus into letting him go to join Hestia Familia on good terms (which is no easy feat in itself), then he discards his oath to never make a magic sword, in order to help win the Game. In just seven days, he forges two magic swords so powerful that Lyu can keep a full third of Apollo's forces occupied with them - not to mention breaking the castle wall. Even after such an output of effort and skill, he helps Bell fight his way to the top of the castle, and then keeps any reinforcements from arriving, so Bell can fight Hyacinthe one on one.
Mikoto manages to lock down a full third of Apollo's forces by casting Futsunomitamaon herself and catching them in it while running and dodging attacks. To elaborate: Mikoto basically went into an Olympic-level sprint over uneven ground without falling once, while singing an operatic aria in a foreign language at the top of her lungs. (Note: only a bare handful of opera singers in history have ever managed to move faster than a walk - usually a few steps at a time - while singing an aria)
While more a temporary ally, Lyu not only joins Team Hestia as member of an outside Familia, risking the ire of the Adventurer's Guild, she uses Welf's two magic swords to take on a full third of Apollo's forces in combat - and uses said swords to break down a huge section of the castle wall. When the two swords shatter from overuse, she draws her own sword - and still keeps a full third of Apollo's forces focused on her!
Bell not only manages to talk Aiz and her friend Tione into training him for the war game - which would be directly forbidden by Loki if she knew about it - he basically spends seven days getting the crap beaten out of him by Aiz and Tione, then goes to the war game with a bare night's rest. He and Welf become Back-to-Back Badasses to fight through the castle, upon which Bell takes on the Champion of Apollo's familia in a one on one duel, which he wins in spite of Hyacinthe having a weapon directly blessed by Apollo himself. Even better, he does it by remembering some of the advice Aiz gave him in their training.
And let's not forget Hestia. She pulls off some magnificent Chessmastering and diplomatic maneuvering: first, by drawing out arguing over details of the War Game long enough for Bell to get his week of training. At the same time she successfully executes a plan to not only see Soma herself, but she negotiates for Lili's release from his Familia in sufficient time for Lili to pull off her undercover mission. It's implied that she actually came up with the master strategy for the War Games herself (although probably after discussion with Hermes, Welf and Mikoto); at least, Bell is fully occupied with training, and Lili goes straight from Soma Familia to the battlefield. But let's not forget the aftermath of the War Game: remember that bit in episode 12 where Hestia unleashed her pigtails - and her divinity - to protect Bell? Well, try turning it up to eleven, only Hestia is very, very angry. She reduces Apollo to a whining, begging mess on the floor with nothing more than sheer force of personality, then ruthlessly makes use of Apollo's previous Exact Words. She permanently disbands his Familia and exiles him from Orario, and takes possession of all of Apollo Familia's money and assets. In that one moment, Hestia Familia goes from being one of the poorest Familias in Orario to one of the wealthiest. Her Adventurers won the battle, but it's Hestia who wins the spoils.
It's more of a Crowning Moment of Sneakiness, but Hermes probably deserves major kudos for getting Level 4, blacklisted adventurer Lyu onto the field at all, under the radar - and the noses - of not only the Apollo Familia but also the Adventurer's Guild.
Lefiya's turn at this was when the flower shaped monsters were rampaging through town. After being impaled through her waist she refuses to leave Ais and the others behind as they didn't have their weapons. She then proceeds to use her magic and copy one of Riveria's spells, freezing the monsters solid and saving the day.
She may be a villain, but Revis stopped Lefiya's spell with her bare hands.
A huge monster suddenly appeared, but working as a group, Loki familia managed to take it down with the other adventurers at the 18th floor barely being able to help out at all. This moment among other things show everyone just why the Loki familia is considered one of the strongest.
Chapter 14 (Episode 5 in the anime) of the Sword Oratoria manga has Finn show why he's a Level 6 and called the Braver by matching and surpassing the murderous Tamer. He only gets a broken finger from a punch he threw at her because of how durable her body was.
In the main series, it was mentioned that Aiz managed to beat a deep-floor floor boss by herself. Here, we get to see it happen, and it's INCREDIBLE! We see every moment, every time Aiz came close to dying only to rise back up with minimal support from Riveria, taking on hordes of mooks summoned by Udaeus and doing massive damage to the floor boss itself before finally achieving victory. She almost died several times, and was bloodied and barely conscious at the end of it, but we see first hand just why Aiz is one of the strongest adventurers in Orario.
The thing that announced Bete, Lefiya and Filvis' entering the pantry at the 24th floor was Lefiya's wild range fire magic, the power behind that shot even scared one of the mage in Hermes Familia.
The battle with the Demi-Spirit in Chapter 39 of the Sword Oratoria manga starts with it unleashing not one, but two wide-scale magic attacks that ultimately decimate the Loki expedition. It broke their spirits, left Riveria and Gareth down for the count, and all seemed hopeless. Then Finn gives a rousing speech that not only gets them back on their feet, but pushes them to face the enemy.
The whole battle at the 59th floor definitely counts. From Lefiya protected them from the Demi-Spirit's lighting, to Gareth slashing through the vines, and with the last drop of her mind power, Lefiya fired a magic to save Ais. And don't forget Tsubaki, who wielded Gareth's axe and Tiona's Urga at the same time.
While searching for information on who kidnapped Anna, Lyuu wipes the floor with the adventurers involved with it.
Lyuu was losing the poker gamer badly against the villain, but Syr stepped in, and with a bit of luck from her lucky rabbit (Bell) and her ability to read people, she completely turned the situation around.
Lyuu manages to beat down the villain, and when he threatens to reveal her identity which would cause problems since she's a blacklisted adventurer, Syr makes it clear to him how bad an idea that is by mentioning who their boss is and which familia she's still part of despite retiring. Mia of the Freya familia.