The first job we saw Cosette in was as a waitress in a pasta place.
Her most recent job before working at Coffee of Doom was at an ice cream parlor. Other employment included Fondue Hut, Mambo Juice, and Video Hut. Every place she worked before Coffee of Doom has gone out of business, some because of legal trouble; Dora was concerned that she might be cursed. However, a witch Dora knows concluded otherwise.
She first appeared in, where it took her apparently a long time to get the nerve to ask Marten out. When he revealed he was already seeing someone, she was devastated.
Unfortunately, she tried to "drown her sorrows" at Coffee of Doom, where she met up with Dora - who was none too happy to find out she'd hit on her boyfriend.
The piece d'resistance of the whole first story arc was when Marten and Dora went out for "makeup pasta", and their waitress was Cosette.
After the rejection by Marten, she later found out about Steve in . She finally met him in . After going out on a date, Cosette ended up in the hospital - due to some rather interesting circumstances.
Cosette's full name was revealed when Steve found her name in the Smif Yearbook.
It should be noted that Cosette's "original" character name was Cosette Beauvais. Her last name was changed to "Hurlbut" in May of 2010 as she started to use Twitter after a party with the rest of the gang at Marten's place - as Jeph forgot that he'd already revealed it.
In retrospect, it makes more sense that Tai would remember Cosette's name, as it's likely they'd be on the same page of the yearbook (though Tai Hubbert would come before Cosette Hurlbut).