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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 healthy now.
Ensemble Darkhorse: Cardbo/Danbo, the cardboard robot, who got an action figure despite having only appeared in two chapters that are far apart, and he 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. He even comes with his own customizable plastic model! BEHOLD!
There is little to no romance in the series itself, but Koiwa X Fuuka is easily the fan favorite. It helps that most of the ship tease that exists in the manga at all are between them.
For some fans, Asagi and Torako who are close friends that hang out a lot.
Hollywood Homely: Fuuka is often unfavorably compared to Asagi, whose beauty is constantly praised and has gotten the attention of Jumbo and Yanda. Yotsuba goes as far as to call Asagi and Fuuka the "pretty sister" and "not-pretty sister" respectively before she remembers their names properly in the 2nd chapter. Though Jumbo does actually call her a Bishoujonote "pretty girl" when he first meets her - he's just also quick to demote her to "just a shoujo" after he meets Asagi.
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. Asagi even tells Tora off once for saying she has fat legs, saying Tora is just too skinny.
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. It helps that the outings we know of, do indeed seem like date spots; an Okinawan beach, a Hot Air Balloon race, and a Firework Display. Its also shown that these outings are always, or at least planned, for just the two of them.
It doesn't help that when Jumbo demands to know who "Tora" is, after Yotsuba draws a picture of Asagi and her, Yotsuba describes her in a way that sounds like Tora is Asagi's boyfriend, much to Jumbo's dismay. It becomes something of a running gag later on.
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 that "they seem to get along together really well". A few chapters later, Koiwai teases Jumbo with a picture of it.
Close friends Asagi x Torako shipping, as seen in the Les Yay entry.
Koiwai and Jumbo's closeness with each other have garnered a few fans for the ship. Koiwai is never seen with other friends (besides Yanda), Yotsuba is convinced Jumbo is his only friend, and some panels with the two of them with Yotsuba make them look like an actual family.
Miura tends to get shipped with Ena as they hang out with each other a lot and no one else their age. She's also more frequently and more disturbingly shipped with Jumbo due to their Sitcom Arch-Nemesis friendship.
There has been some Yanda x teenaged Yotsuba, since if she was older their relationship would have all the makings of a Slap-Slap-Kiss romance.
Miura has Boyish Short Hair and wears boyish clothes, so many readers mistake her for a boy in her first appearance. Even Jumbo has to ask if she's a boy or girl not long after first meeting her.
Torako also has short hair and doesn't dress femininely so she can confuse readers too. The only physical hints of her gender are her chest and the slight eyelashes found on most of the female characters.
Koiwai has shaggy shoulder-length hair and his facial features are on the softer side, so readers often mistake him to be female until they see Yotsuba calling him Daddy in the first chapter.