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.
Tabitha's long lost twin sister, who seems to have been created from a checklist of Mary Sue cliches.
A common complaint leveled at the elves. They are more in tune with nature, are not canonically affected by the impending disaster in the novels and have better tech than anyone who does not come from earth.
Colbert Bump - The series saw a huge upswing in popularity on the Spacebattles Creative Writing board after The Hill of Swords showed up on the Fanfiction Recommendations thread. Prior to that, it had been an almost complete unknown in the community, now SB is effectively the epicenter of the ZnT fanfiction community.
Siesta treats Saito much better than Louise, but is loathed by fandom for that. Part of the reason is flanderization in the anime.
Louise suffers from the people who like Siesta over her or Kirche or Tiffania or Henrietta and Tabitha. Louise isn't popular among the fans of the others in the female cast.
Dismissed Gender / Double Standard / Values Dissonance - Invoked plenty of times to say the least. Louise is not acting different from what nobles did during Ancient Rome and Middle Ages. At one point killing a commoner (not to mention beating them up if they displeased you) was not a crime at all (that usually didn't apply to townsfolk, who were under protection of law to the extent that you had to have a valid reason to kill them). One has to wonder if the series would be successful if Louise were male...
Macekre: The anime devolved from grey-coloured story with occasional breathers to Tsundere comedy with a lot of Fanservice and something happening in the background. Because of that, some things were slightly changed, and some cut off completely, often resulting in removing heart-warming or awesome moments just to put more female body parts in (a very prominent example being Saito's death, which was the reason why Louise's company had come into contact with Tiffania, and an important part of Louise's character development, being changed into Saito coming back a short while later on his own, and then dragging Louise to Westwoodfor no apparent reason; another fine example is Agnes' behaviour towards Colbert after his faux death, once again an important part of someone's character development (in this case, of two characters, even), being cut completely).
"Tabitha has taken over Siesta as my favorite character, and not just because she is channeling Yuki. She is made of pure win, I don't usually like the stoic silent types, but something about Tabitha just makes my inner moe meter go berserk."
Louise counts when she's in dere-dere mode.
Moral Event Horizon: There are fans that believe Louise crossed this the first time she whipped/exploded Saito or the Nth time afterward.
OT3 - Louise/Saito/Siesta is a popular pairing. It's likely because Louise and Siesta balance each other out. This was teased a little in the 12th volume where Louise says she would be disappointed if Siesta gave up on winning Saito so easily.
The anime works at a conversion ratio of three to four volumes per twelve episodes. Due to this, at least two thirds of the source material has been outright omitted, and the rest truncated or chopped up. Though the manga and novels aren't immune to this criticism.
Wangst: The author tries to justify Louise's abuse by showing her poor self esteem but most people don't see that as a good excuse to beat someone.
You really can't help but feel sorry for Saito because of the abuse heaped upon him by Louise. Hell, even the other familiars feel sorry for Saito, and see his treatment as both a familiar and a human as horrible.
Tabitha qualifies, once you learn of what happened to her parents and what her uncle does to her.
Jerkass Woobie: Louise in the light novels if you can look past her treatment of Saito. She has a HUGE insecurity complex, a horrible school life, and her treatment at home is only better because of one of her elder sisters.