Nichijou/Fridge

Fridge Brilliance:

In episode two, Mai cheats at the stairs-climbing game by recalling a "spell of resurrection". It's later revealed this is a password for Dragon Quest, and that what she enters is incorrect. She was making an obscure video game reference that very few people would get (never mind her friends), and she's been getting it wrong all this time. Good thing she wasn't with another hardcore fan.
- Obscure in the West. In Japan, Dragon Quest is more popular than most religions.
"First comes rock, then comes paper / You can't throw all of them..." Mai tries to in episode five and scores a Critical Hit. Then again, it might be because most people don't have a solid wooden statue handy.
An interesting detail when you pay attention is that, while segments featuring the trio of Mio, Yukko, and Mai are frequently sprinkled with What Do You Mean It's Not Awesome?, the segments with the mad scientist, wind-up robot girl, and talking cat are far more mundane.
In episode six, it is revealed that Nano is deathly afraid of cockroaches. It kinda makes sense that a robot wouldn't like bugs.
- Nano is shown to be afraid of thunder in episode 13. She's a robot. What normally comes along with thunder?
In episode 12, a coffee shop employee told Sakurai that she'd have to tell her which sweets she wished to buy, despite the fact that she was very clearly pointing at them. Why? She's blind.
- Jossed. It turns she's not good explaining things to people. See episode 16 where she didn't help Yuko explain what were those weird coffee size names.
- Plus if you've ever worked in food service with a similar setup, you know how difficult it can be to tell what someone is pointing at when they're standing on the other side of a display case. Much easier when they just tell you the name.
This is half-Fridge Brilliance half-Wild Mass Guess: The Professor is afraid of dogs because she's scared they may bite her. Yet, her favorite animal is the shark, which is known for biting. This seems contradictory, but consider that the Professor is a scientist. A dog might bite her, a shark will bite her. It's not the dogs themselves she fears, it's uncertainty, the unknown variable.
- You may be giving her too much credit for logical thinking.
Nano desperately wants the Professor to remove the key from her back. However, the Professor always refuses to do so--not because she wants Nano to suffer for the sake of cuteness, but because she wants Nano to be accepted by others for who she really is.
Considering how much landmass Japan actually has, it's not surprising for landowners to get a somewhat inflated attitude over their own place in society, which pretty much explains Sasahara.
The toast Helvetica Standard in Ep.5, when the toast hits the ceiling the lamp shook, and this lamp looks exactly the same as the one that fell on Mio and Yuuko's card tower much, much later.

Fridge Logic:

If Mio immediately regretted drawing Sasahara in front of Sakurai and Yuuko, how did it end up on the quiz? It would have to have ended up in Sakurai's possession in the end, and obviously Mio wouldn't let that happen.
How do you get hair through wooden cubes...?
- There's holes in them?
  - But there aren't, at least drawn. And in Yukko's dream sequence they're able to be put on via vertical insertion.
    - Two halves, hinged, with a clasp or magnet on the other side.
I know it's a comedy cartoon and logic means nothing, but I can't help but wonder how Nano gets her shirts on through that big wind-up key. There's no visible seam or buttons in the backs and we never see her in the action of putting anything on over the key. Then again, we are talking about the creation of someone who could make a cat talk with only a scarf. Said eight year old was actually the one who presented Nano with her school uniform, so maybe she had to modify it first?
- As of episode 24 we finally see the answer, there are buttons along the back of her shirt underneath the key. This info brought to you courtesy of the Science Teacher spying on her changing for gym.
If you look at the book Nano is reading in episode 11, the title is "Dostoyevsky". While it's certainly unclear which work would that be, does this add anything to her smarts?

Fridge Horror:

The Helvitica Standard in Ep.18 stands out in all of them for being very different. Not only is there a beautiful Art Shift, but the tone is also quite sentimental and nostalgic. When the book the girl was looking for turned out to be emptied out and was hiding a gun, this initially seems like a Crowning Moment of Funny. However, this scene can easily be interpreted differently that turns it into a massive Funny Aneurysm Moment. The girl did not seem surprised when she found the book, and in fact, she stated that she wanted this specific book, not any other copy. If she did indeed know there was a gun in this book, what do you think she meant by "it brought colour into my grey childhood"?