These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Sola killing Kayneth. While he was a Jerk Ass, the fact that Sola murders him after he gave her his Command Seals and that she murders him willingly (no curse that excuses her craziness) without any remorse whatsoever doesn't make her terribly sympathetic.
Kirei and Wade stopped being funny when they killed Illya
Illya's final thoughts are absolutely heartbreaking: "Was I really like a flame Berserker? Was I really beautiful before I burned out?" then TIM drives the knife in even deeper with her internally begging her dead mother and father to show up and save her as she dies.
Unfortunate Implication: some fans felt like Sola-Ui's affection for Lancer (who in this fic is a teenager) sounded more like attraction than maternal affection, giving him some kind of pedophile subtext. It doesn't help that in canon, Sola did fall in love with Lancer (who was an adult then). The Infamous Man denied such subtext however, and insisted that their relationship is purely mother/son-like.
Its possible that Lancer dating Fuji in the sequel was meant to reassure readers about this.
Villain Sue: Kotomine wants to become this after his Fourth Wall broke.
Wangst: Caster's death. She died a complete Hypocrite blaming people for things that they had nothing to do with and calling them selfish for wanting the Holy Grail while being completely selfish herself because she also wanted the Holy Grail.
The Woobie: Archer; poor guy wants nothing but to solve the Holy Grail War without anybody getting killed. Which is pretty much impossible when you consider you are in the Nasuverse, in a conflict where it's required to kill other Servants for it to be completed. He fails misterably to save Caster and sees him die in front of him, before having his Master betrayed and killed while he is taken over by the Big Bad and sent with the order to kill Saber. In the end, he has to sacrifice himself to not kill anybody.
Avenger, in the sequel, is just as bad as Archer was, if not worse. It's worse because you eventually realize that this EMIYA is not the EMIYA we've known from canon. This is Shirou from Heaven's Feel where he killed Sakura and walked the path of a Hero instead of saving her and giving up that path. As his backstory is slowly revealed, you can't help but feel sorry for him for putting up with all that he did. To be a Hero, he became the Villain of the World so that people focused their problems onto him, similar to what happened to Medea in canon. In the end, he felt it was worth it, so long as it brought the world peace.