Abusive Parents: Kenny's parents, per usual. This time they're being arrested not only for fighting but also for the meth lab.
Mr. and Mrs. Weatherhead also count.
Adults Are Useless: Subverted. Mr. Adams is a dimwitted man who tells Penn State jokes more often than he should be helping children, but when he sees how bad the Weatherheads are, he is very ashamed for having put them there, which is more than can be said about the other adults on the show. He does put the children back into their original homes, but only because he felt the system couldn't help them.
Apple of Discord: Pabst Blue Ribbon, which of course label whoever drinks it as white trash. Mysterion uses it to his advantage to bring down the already-abusive Weatherheads.
Berserk Button: A blink if you miss moment, but when Mr. Weatherhead is preparing to take the kids to the "PunishmentRoom" for talking about Karen's Guardian Angel (Mysterion). Kenny immediately shoots a Death Glare at Mr. Weatherhead Before suiting up to take him and his wife down.
Big Brother Instinct: Kenny, as Mysterion, jumps to Karen's defense when she's being bullied by Jessica Pinkerton at their new school.
Expy: Mr. Adams is closely based on Due Date's Ethan Tremblay, who regularly thinks of himself as a comedian and gives his audience signed photographs of himself.
Mr. and Mrs. Weatherhead are both a parody of highly religious and abusive foster parents seen in the narrative cliché and an exaggerated caricature of agnosticism.
Laser-Guided Karma: Cartman frames his mother for running a meth lab, leading to her arrest. Near the end of the episode, Cartman himself is arrested for filing a false police report. This episode wasn't the first time both of them got arrested.
Mr. Exposition: Butters reveals to have some computer knowledge when he helps Cartman research who's the poorest kid in school after Kenny.
Shout-Out: White Trash in Trouble parodies programs like Cops and Campus PD.
You Are Not Alone: Mysterion assures Karen of this on the first night that they're at the foster home.
Your Mom: Cartman keeps telling 'Yo Mama so poor' jokes throughout the episode, most of which are aimed at himself due to being the poorest kid in school (or at least in 4th grade) after Kenny. Once he finds out that there's an even poorer kid in the new school near the foster home, Cartman targets him instead, resulting in the following:
Principal: "A 20-minute song and dance number featured with 47 'Yo Mama So Poor' jokes directed at Jacob Hallery, which ended in a finale with fireworks."