As the younger sister, she is commonly called 「妹様」 or "imouto-sama" by fans.
A common meme regarding Flandre is 「フランちゃんうふふ」 ("flan-chan ufufu") since many people attribute the slightly sinister laugh "ufufu" to Flandre, as well as the fact that her name is commonly affectionately abbreviated 「フランちゃん」 ("Flan-chan").
In some fanworks, the crystals on her wings are sometimes shown to be storage for Magical Power.
In many fanworks, Flandre is shown with one of two different personalities, or possibly a bipolar one. One is a personality based on her appearance, been very childish, and the other is a super sadistic and violent mass-murder-like personality, based on her possibly accumulated hate of the years trapped in the mansion.
Her boss theme ("U.N.Owen was her?"), has become popular on various video websites because of videos such as this and this and has been the first contact with Touhou for thousands of internet users.
Flandre has been depicted in some fan-art alongside Koishi. This could stem from the fact that they both are extra stage bosses and possess destructive powers greater than those of their sisters, and that they (Satori and Remilia) both own mansions, Remilia the Scarlet Devil Mansion and Satori the Earth Spirit Palace. Not only that, but Flandre and Koishi have spent most of their lives in their sisters' mansions.
Because she is portrayed to be childish, there are some music videos of her with Cirno.
Some fan works also depict Flandre as friends with Marisa, though the exact reason is unknown (though possibly because Marisa agreed to 'play' with her)
Flandre's ability is "destruction of anything by crushing the eye" : this ability is seen in fanworks when her fist is clenched (certainly crushing "the eye" of the thing she wants to destroy). So far we can see her destroy : the sun (like this), Marisa's Mini-Hakkero (like this)and even a meteor (like this).