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.
Angst? What Angst?: Yotsuba was either orphaned or abandoned overseas, then taken in and raised by Dad. At least partly justified, since Yotsuba seems to have been too young to remember her real parents, and anyway, what matters to this story is that she's happy and well-adjusted normal healthy now.
Ensemble Darkhorse: Cardbo, the cardboard robot, who got an action figure despite having (at the time) appeared only once, and has even ascended to the noble position of Official Mascot of Amazon.co.jp. Just don't tell Yotsuba it's Miura in a costume made for a school assignment.
Now with his own customizable plastic model! BEHOLD!
Fan-Preferred Couple: There is little to no romance in the series itself but Koiwa X Fuuka is easily the fan favorite. Rule 34 is generally frowned upon when it comes to this series but they obviously exist, and probably 90% of them are on these two. It helps that most of the ship tease that exists in the manga at all are between them.
Fan Nickname: Yotsuba appears on 4chan's 404 Not Found page, and is thus known as "404 Girl". A recurring character that Yotsuba sometimes plays with is called "Dotty" by the fans, after her eyes being always drawn as 2 dots.note Was unnamed for a good while, but was eventually given the nickname "Mi-chan" in the chapter "Yotsuba & the Sandpit".
Hollywood Pudgy: Fuuka is frequently stated to have fat legs, and is even teased about this by Asagi and their mother. Look at a picture of Fuuka alongside any other teen girl character in the series, and see if there's any major difference in their body types.
Then again, Asagi and mom are jerkasses at times, so they could just be messing with her.
Les Yay: Asagi is never seen socializing with anybody outside of family and her neighbors—except for Torako. Asagi and Torako also love to go on long outings together, and some people feel those show all the telltale signs of actual dates.
Memetic Mutation: Yotsuba is used as a mascot by some Image Boards, most prominently 4chan, which uses a clover as its logo and runs on a modification of Futaba called Yotsuba.
In addition, Memetic Mutation is exemplified within the series by the teddy bear Duralumin, who shares a name with a metal alloy but is really Yotsuba's mispronunciation of Ena's stuffed bear Giulietta.
Sacred Cow: Don't even mention anything that involves Rule 34 images of Yotsuba of any sort to Anonymous.
Shipping: A popular ship is Koiwai x Fuuka, prompted by their multiple interactions (including classmates asking Fuuka if Koiwai is her "husband") and hinted at in Azuma's proto-comic for Yotsuba&!, "Try! Try! Try!". The multiple blush-inducing situations that Fuuka experiences around Koiwai (including the paper-cup telephone conversation and a comment in Chapter 67 about how her eyebrows are cute as they are) do not help this. That Azuma in a side comment was "scared" by Fuuka's classmates finding Koiwai attractive hasn't deterred shippers.
In chapter 61, Koiwai gives Yotsuba a "helicopter ride" (swinging her around by the arms so her feet leave the ground). Then Asagi lines up to try it, and seeing them circling each other hand in hand, Torako comments "They seem to get along together really well"... Ooh, Azuma-sensei, you ship-teasingdevil you...
A few chapters later, Koiwai teases Jumbo with a picture of it.
Surprisingly enough, moments exist even in this most feel-good of series: In the first Festival Episode, immediately after being told to stick together, Yotsuba lets go of her Dad's hand to run up to a cotton candy stand, thus losing him in the crowd. You watch a five-year-old girl look desperately around for her father, gradually getting more upset until she's crying and shouting "My name is Koiwai Yotsuba! My name is Koiwai Yotsuba!" at the top of her lungs, and try not to tear up.
When Duralumin is broken. Can Asagi save Duralumin's voice? Or will she never go "Maaaaaa" again?