Elan had Nale; over the course of the fifth book, the position was usurped by Tarquin, cemented when Tarquin killed Nale.
Haley has Crystal (even if she has no idea what an arch-nemesis is), Bozzok, and Sabine.
The main female in every evil joint the Order encounters is bound to become an Arch Enemy of Haley (the Thieves' Guild - Crystal, the bandits - Samantha, the Linear Guild - Sabine and Kitty the hawk, Team Evil - Tsukiko), though only about half of them stick around. The exception so far is the Empress of Blood, probably because no one wants to be the Arch Enemy of that.
Belkar, at least, is of the impression that Sabine is Haley's one true Arch Enemy. This is actually Lampshaded by Haley, who notes this tendency and comments that she should have chosen 'Favored Enemy - Flying Skank' as her class ability.
Sluggy Freelance: Bun-Bun's arch-nemesis is Santa Claus, whom he tries to kill every Christmas that he's not stuck in an endless void outside of time. Bun-Bun doesn't actually remember why he hates Santa so much, but he's not about to let that stand in the way of a good grudge.
Dan and Mab's Furry Adventures : Until the latest chapter the series really wasn't that melodramatic, but now for Dan it's a toss up between Dark Pegasus and Regina Darkblood. Dark Pegasus is more dangerious and re-occuring (Dan has killed him twice), but with Regina it is far, far more personal.
Lightbringer: Darkbringer for Lightbringer. Both believe in an objective morality, but whereas Lightbringer believes what is "good" is the correct moral standard Darkbringer believes what is "evil" is the correct moral standard.
In Spinnerette, Spinnerette's arch nemesis is... Spinnerette. No, this isn't a case of Enemy Within, they just both happened to pick the same supranym. And then started fighting over it. They now distinguish each other as "Good Spinnerette" and "Evil Spinnerette."
In Homestuck, the trolls recognize archenmity as a type of romantic, sexual relationship known as kismeissitude. Examples are Snowman/Spades Slick, past Eridan/Vriska, past but very recent Terezi/Vriska, and Karkat/John (unrequited because human John doesn't have a concept of romantic hate). In Act 6, Karkat seems to be soliciting Dave as a kismesis, as Dave has drawn Terezi's romantic (non-hate) attention away from Karkat. Dave does not react well to this; like John, he's not comfortable with romantic hate, especially from another male.