Cirno's catchphrase in doujin works, "I'm the strongest!", comes from her dialogue in PoFV, in which she announces this in her scenario, while being everyone else's stage one boss. She also says it when winning against herself in Phantasmagoria of Flower View and in Hisoutensoku.
Because of this, some people tend to call her "the self-proclaimed Chuck Norris" of Touhou.
Despite not appearing in most Touhou games, Cirno is one of the most popular characters mainly due to fandom making references to her in non-related topics.
She is shown often with a form with control over fire.
Cirno is always shown working together with Letty in most doujins, fan-arts, and others despite showing dislike for Letty. Fans most probably did this because of their similar powers of controlling the Cold.
Some artists like to show Cirno as being very smart. Sometimes it's played straight, but it's most commonly shown as accidental genius. For example, one comic shows her getting every math problem on a chalkboard wrong...until you do the math in Base 9 (which is a bit tougher than the base 10 we normally use), in which every answer is correct. Almost always played for laughs.