Among the Touhou fandom, Suwako's hat has almost become an entity of its own, largely thanks to the imposing leer of the eyes on top. Its most common name is Pyonta.
Suwako is heavily symbolized as a frog. Cirno freezes frogs. The results in fanon are obvious. The frog jokes include relationships with Super Mario Bros. 3 and the Hypnotoad.
Because of that, Suwako is commonly called 「ケロちゃん」 or "Kero-chan," which appears in one of her spell cards, 「ケロちゃん風雨に負けず」 which is "Kero-chan Braves the Wind and Rain." This nickname was mentioned, for example, in Tetla Pot's PV of Silver Forests's song, Kero⑨Destiny.
Although Nue is the first character officially drawn with Zettai Ryouiki, Suwako has almost always been drawn this way in fanart, possibly because of her sprite in Hisoutensoku.
Fanart has shown Suwako constantly obsessing over Cirno's taste. This could possible be because Cirno has a hobby of freezing Suwako's favorite animal, the frog. Even music videos for Touhou has one where Suwako sings about how delicious Cirno seems.
Since frogs are pictured coming out of her dress, some fan art has explored the idea that she gives birth to those frogs via eggs that bud from her back. This actually occurs with the Surinam toad species.