Axl's Piercing Parlor is the piercing and tattoo shop in Lawndale on Dega Street run by Axl. Seen in the episode, "Pierce Me", it is infamous for its hygienic conditions and for its not very strict compliance to the letter of the law. (Axl asks Daria if she's 18 before piercing her, but takes Trent's word that she is, despite Daria herself not confirming the fact.
"The Daria Database" tells us how lax the health and safety standards are, and that health insurance companies refuse to cover you if you get injured or ill from using Axl's.
Burnout Girl may be living above it.
While only shown once in canon, it has been used with some regularity in fanfiction and in some cases, expanded. For example, Trent moves into a tiny apartment behind Axl's in the "John Lane" series.