Apparently this page (images on page defunct; archived here) is where the "Suppatenko" (literally, "Naked Heaven-Fox") trend started. [NSFW; Not recommended for pure Ran fans.] Since then, in many fan works, she's been pictured as a flasher.
She's usually portrayed as a motherly figure, especially towards Chen.
In many fanworks Ran is portrayed as highly-excessive nosebleeding whenever she sees Chen.
According to Ran's spellcard, "Princess Tenko", and the phrase on the first scene of EX stage of PCB, "A mysterious occurrence on the crossing to three countries", some fans of Ran think her as Tamamo no Mae, rather than White-faced Golden Fox of Nine Tails, or that being showed above.
Ran is often depicted as Yukari's "voice of reason" in fan comics.
For some reason, she is depicted taller than most characters in fan works. However, According to ZUN, Ran is not taller than Yukari. Yukari is one of 3 tallest characters in the entirety of Gensokyo.