Crime fiction is about breaking the law: who broke it, how they broke it, how they got caught (assuming they do get caught), and what happened to the victims of their misdeeds. This literary genre is more common in Daria fanfiction than one would think. Murder and kidnapping are often seen, and quite often the perpetrators are Daria characters (see Evil Daria). These stories very rarely cross into the adventure category, and crime stories with strong horror, fantasy, or science-fiction overtones are discussed elsewhere. Stories about Daria supervillains are described in the section about superheroes.
Crime in Daria Canon
People in Lawndale commit crimes a lot: Angela Li is implied to be bending various rules to carry out her reign of terror, the bursar at Middleton College is a 'legitimate businessman', Tokyo Toby violates health rules, Upchuck is a creepy peeper, Helen's firm helps people skirt the law etc. There's the entire Quinn subplot in "Fire!" too, where she's pursued by a stalker who steals and hacks to carry his works out.
However, there's only two episodes that end up being a crime story as it usually seen: "Arts 'N Crass" and "Murder, She Snored". The latter is a murder mystery taking place within a dream, so the murder won't disrupt the series. "Arts 'N Crass" is entirely in continuity though: Daria and Jane commit a crime because the system fails them, their motives are thoroughly examined, there's no question they are guilty, and they escape punishment. As this is in continuity, the crime in question is petty vandalism. ("Speedtrapped" has characters in jail but as a comedy B-plot rather than the result of proper crime)
Crime in Daria Fanworks
As noted earlier, crime fiction in this fandom can have any sort of perpetrator, even normally "good" characters like Daria. A vital issue here is the question of whether any characters who appeared on Daria could be considered criminals, if not thoroughly evil. Bobby Stuart is a good example of a criminal. One PPMB thread explored this topic and produced a long list of potential suspects for fanfiction ("Irredeemable Daria Characters"). See also Evil Daria and Thom.