A subtle moment in volume 1 that's significant because it's before Haseo's character development: Bo comes to the shop looking for an item for Saku that Haseo says is worth 6000GP. Haseo initially scolds Bo for not having enough money until Bo mentions it's a birthday gift for Saku, at which point Haseo offers him the item at a discount without question or hesitation, and sounds genuine when he says "Give my regards to your sister". Turns into funny when Atoli comes up afterwards.
Haseo's relationship with Silabus and Gaspard in general. Initially, Haseo thinks of his Canard guildmates as annoying pests, but as the games progress, he comes to be extremely protective of them, and they in turn look up to and respect him. Their Friendship Moment during job extension quest in Reminisce in particular springs to mind.
When Haseo runs after Atoli in the final main-storyline cutscene, her expression upon seeing him is priceless.
Shino returning and asking both Haseo AND Atoli to come. Atoli isn't sure at first how to really react despite knowing about Shino and runs off after an awkward moment in tears. Haseo isn't sure if he should follow her at first, but Shino encourages him to go after her because she knows that Atoli has become very important to him. Despite the fact that she developed feelings for Haseo too, she's perfectly willing to let him go if that's what he really wants.
In Vol. 3 when Alkaid returns and Haseo hugs her sobbing. Call it narmy all you want, but that scene is almost granted to leave you with a warm fuzzy feeling.
When Aura refers to the Infinity Eight as "Little Children of Morganna." Given how much damage Morganna did to The World in general, and how much grief she caused Aura in particular, this simple gesture of acceptance and forgiveness speaks volumes.
After three games of denying that Antares is his teacher, Haseo finally calls Antares "Master" in the post-game email wrap-ups at the end of Redemption.
It also happens in Reminisce after the Job Extension quest when the Friendship Moment inadvertently leads to Gaspard revealing Antares set them up for it to teach Haseo another lesson. Haseo, after a bit of resistence, finally gets the lesson and quietly calls him "Master" without anyone hearing.
For years, the final farewell event of The World was a non-playable event only appearing in the adaptations. Volume 4 is set in those last days, allowing you to bear witness to it as a player would have seen it.