Angst? What Angst?: The Main Characters and other former members of the Debauchery Tea Party take their unexpected imprisonment in a video game surprisingly well, and complain that most of the other players are getting too worked up over it. There are many players who angst about it, but those players are not the main characters. It helps that their lives are not directly threatened; they respawn when killed. The players are more concerned about incertitude and social troubles than surviving.
The episodes/arcs that revolve around the junior members of the Log Horizon guild are found to be quite fatiguing for some people, and want the spotlight to move on from what they see as less interesting characters to follow along with, and would rather see more of Shiroe's adventures, or perhaps pass the spotlight onto characters that they've felt have gotten the short end-of-the-stick in screen-time such as Nyanta and Crusty. Adding to this, Minori suffers from major Die for Our Ship regarding her ''crush'' on Shiroe.
As mentioned in Broken Base, Akatsuki's character arc in Season 2 was this for some because they think it dragged on unnecessarily long.
The animation studio switch from Satelight (season one) to Studio DEEN (season two) split the fan-base as to whether it was an overall upgrade or downgrade. You may even be in the third camp stating that nothing feels different.
Akatsuki's character arc in season two where she goes after the Christmas killer is very hit-and-miss amongst the fans. Some found that the arc didn't need to drag on for four episodes, and were getting impatient, because what they truly wanted to see was the action going on in Shiroe's raid on Elder Tail's gold stash. Others were fine with it, stating that this is what Log Horizon is best known for building up the problem before finally dealing with it, and eventually got to what everyone was begging for; an action-packed final episode of the arc dealing with the murderer. Akatsuki's Character Development seen throughout this arc is also a huge plus for some people as it was something that was constantly asked for all throughout season one due to her popular status.
William's speech is very divisive among the western fanbase. Some see it as a waste of time, others find it very relatable and loved it. It's very much a cultural issue, as the Japanese fanbase absolutely adores William for it.
Ensemble Dark Horse: There are several minor characters who have become big hits in the fanbase.
After her debut in the anime, Serara is highly popular among viewers and she's not even a member of Log Horizon.
Nureha, one of the rivals for Shiroe's affection, has a significant following among Japanese readers, though for an entirelydifferent reason. She's the one with the Courtesan (娼姫, literally "whore princess") subclass, by the way.
Isaac, one of the guildmasters, popularity exploded, at least amongst the Japanese fandom, after the release of the Black Knight vs. Berserker Drama CD in volume 7's special edition.
Silver Sword has a large fan following despite being prominent in only one volume.
Much of the beginning of Season 2 involves Shiroe trying to figure out how to generate enough funds to pay for the Round Table's upkeep costs for maintaining Akihabara's infrastructure. He then comes across an idea to exploit a hole in the game's logic: mainly, if monsters drop gold, where does the gold come from? His investigation takes him to an underground raid zone where it is rumored that it holds an endless supply of gold. So basically, Shiroe is looking for the fabled Loot Cave.
This franchise takes the Elder Scroll series as one of inspirations, then comes the Elder Scroll Online at April 2014. There was a guild named Log Horizon there to boot.
Hollywood Pudgy: Averted in Kyouko's case, as the way she's drawn she actually does look pudgy, especially when compared to the rest of her guildmates, who are almost universally slim.
Moe: Akatsuki is darling and huggable. It's acknowledged in-universe.
Seasonal Rot: Season 2 is this according to some fans. It's mainly an issue of personal preference as the story's focus is much wider, breaking up into largely separate plots that spotlight different characters. There's also the aforementioned Broken Base over the DEEN's animation quality.
Shipping: Mostly avoided. Much like in his previous work, Maoyuu, Mamare prefers One True Pairing, as evidenced by the front and back covers of book 5 (which showcases the more obvious pairings). However, he has acknowledged that he conceptualized three possible love interests for Shiroe, which obviously has gotten the shipping-inclined among the fandom very interested on who's the "best" girl for him.
Some fans have complained about Shiroe taking significant damage in their first battle, instead of only trivial amounts in the novel... even though his dwindling health is only used as a cliffhanger for the In Medias Res opening scene, and isn't brought up when the full battle is shown.
Some of the censorship in the anime, such as the lack of blood, cutting off Naotsugu's perverted lines (courtesy of Akatsuki's flying knees) as a running gag, and hiding Nureha's backstory have caused complaints about the show. However, this is justified since the show airs on NHK, which is a publicly funded network that caters to children.
Readers of Sword Art Online have claimed that a major plot twist of Log Horizon ( the NPC population is sentient) was also in Sword Art Online. It's really a Cryptic Background Reference, but cited on a few occassions (once in the anime) as a reason why Kirito dislikes having innocent NPC's killed during Boss Raids. Because SAO is mostly a single-person focused story, the matter isn't explored beyond Yui's involvement (and Fina to a lesser degree). Again though: TRON and Neuromancer did it first... so it's silly to nitpick.
The premise of an MMO being real, with humans having a profound impact on it (and vice versa) was the ambiguous ending to the game Star Ocean: Till the End of Time.