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.
Badass Decay: Nyaro takes this to the logical extreme. At the beginning of the episode, he is introduced after having slain several gods, with bad lighting that barely reveals his features and apparently reveling in the carnage. By the end of the episode, he's been chased by the police, defeated several times by accident, and his One-Winged Angel form blasts off after basically having his toe stubbed. He actually puts up some of a fight in the light novels, but Nyarko still hands him his ass and sends him blasting off at the end.
Crack Pairing: Considering the main pair is a human and an Outer God, you'd think this be enough. But there's also Cthugha x Nyarlathotep. In the regular Mythos, one was used to fight off the other; in Nyarko, this is acknowledged as the two races being bitter enemies. On top of that, Nyarko is utterly repulsed by Cuuko's advances, and she originally threatened to kill Mahiro, so all told Nyarko has some pretty solid reasons to not wanting Cuuko around.
Considering the official explanation that aliens love Earth culture, all of the show's copious references can easily be explained as the alien characters being total Otaku who love to pepper their daily conversations with lines from their favorite shows — just like Nyarko does.
A rare example of this happening in the theme song, specifically, the ending of the first season, "Zutto Be With You". Nearly every single line ends with the sound "Oo". Try singing it out loud. Your face will look like you're desperately trying to blow someone a kiss.
Fridge Brilliance or Fridge Horror: The gods of the Cthulhu mythos love Earth's anime, manga, and games - especially, apparently, the hentai stuff based on the doujin Nyaruko was buying in episode one. What's hentai infamous for? Of course it would appeal to them!
Fridge Horror: In the Dating Sim episode, Mahiro is unwillingly jumped around between the key events of the game, skipping over large portions of the school year. We're never outright told how the others experience it, but the fact that Nyarko had the time to knit a (rather long) scarf for Mahiro's Christmas present implies they lived out the whole thing. That in and of itself is pretty scary, but then one has to wonder what happened to Mahiro in the meantime — did he simply disappear, or was he on "autopilot" like Adam Sandler's character in Click when he fast-forwarded time?
Fridge Logic: Acknowledged in one of the Nyaruani shorts. Mahiro is reading the first light novel, and asks Cuuko some questions: she joined Nodens in order to see Nyarko again, and she got into the PDO thereafter thanks to an uncle who works in HR. If that's the case, he asks, why didn't she just join the PDO to begin with and not have to become a criminal? Cuuko responds by lighting Mahiro's book on fire, telling him it's a warning.
"Funny Aneurysm" Moment: In the first season, Nyarko gives Mahiro a Yes/No pillow that depicts her wearing black lingerie; the first time she actually wears that lingerie is during her off-mentioned emotional breakdown and Anguished Declaration of Love in the second season, where it's anything but sexy.
On top of that, her breakdown is caused by the fact that Mahiro isn't willing to definitively say whether he's attracted to Nyarko or not — in other words, because won't just say "Yes" or "No".
Genius Bonus: In one episode, Nyarko performs Eiji Hino's transformation pose; the sharp-eyed viewer (and one familiar with Kamen Rider OOO) will notice that the energy rings that appear correspond to the Kuwagata (Stag Beetle), Tora (Tiger), and Condor Medals. Applying the naming convention used for OOO's forms, one gets Ku-To-Ru - the Japanese pronunciation for "Cthulhu".
Germans Love David Hasselhoff: Nyarko and Cuuko explain that this trope applies on a galactic scale; the conversation was triggered when Mahiro recognized Cuuko's manga as one that only lasted six issues, but the girls explain that it was revived by an alien publisher.
Harsher in Hindsight: Or rather Creepier in Hindsight. Once you see Shantak-kun's human form, it becomes much harder to laugh at jokes involving its eggs, or how it snuck a sausage which was apparently laced with aphrodisiacs in Remember my Love(craft-sensei).
In episode 1 of the second season, Mahiro adopts the Madness Mantra "It's just ink...it's just ink..." as Nyarko slaughters a Nightgaunt off-screen; jump ahead to episode 11 of Nyarko-San W, where the cast goes to a haunted house, and the mysterious stains on the carpet actually are ink.
Thanks to fanart, Mahiro has also been considered this.
Memetic Mutation: The Strange Circumstances of one Mahiro Yasaka, as written by H.P. Lovecr/a/ft, from the /a/ board of 4chan, which is a re-imagining of the events of the episodes of the anime as if they were written and narrated by Lovecraft himself:
The Scrappy: Mahiro often drew a lot of hate from the fans due to his treatment of his three companions.
Cuune. Not a lot of people find "incestuous Psycho Lesbian" very funny.
Hasuta is disliked by some for being a perceived The Generic Guy who only exists to please trap fans.
They Wasted a Perfectly Good Plot: Not exactly plot, but more than a few fans feel that the Lovecraft references should have been better utilized. Some have even accused them of just being window dressing to make the series seem less generic.