Continuity mistake: Peggy Hill's mother seems to change over the course of the show. Prior to the ninth season, whenever Peggy's mother is seen (in the present or in flashbacks), she is portrayed as an older version of Peggy. In the ninth season's premiere episode, Peggy's mother is a lean, leather-skinned rancher who disdains Peggy and lives on a secluded ranch. The change in facial features and appearance is too much to just be a change in style as well.
Continuity mistake: In several episodes, the color of Alamo Beer that Hank, Dale, Boomhauer, and Bill drink, changes color. In "Uh-Oh Canada" the beer is yellow, but in "Death of a Propane Salesman" when Hank pours the beer into a glass, it is brown.
Continuity mistake: Throughout the first couple of seasons, Hank's first truck switches between automatic and manual transmissions. This is most obvious in the episodes "Chasing Bobby", in which it has a manual transmission and "Texas City Twister", in which it has an automatic transmission.
Continuity mistake: The ages of Hank, Peggy, and Didi (Hank's stepmother) are inconsistent throughout the show. In the season one episode "Shins of the Father", Hank states that Didi went to kindergarten with him, so they are the same age. In the season four episode "Peggy Hill: The Decline and Fall", Hank mentions that Didi is three years older than Peggy. Therefore Hank is also three years older than Peggy, if he and Didi were in the same kindergarten class. Finally, in the season five episode "Luanne Virgin 2.0", Hank says that he and Peggy were already married when they were eighteen. This is not possible if he and Didi are the same age and went to kindergarten together. He would have had to marry a fifteen-year-old Peggy if he was married by age eighteen.
Continuity mistake: In earlier episodes, Luanne is very knowledgeable about cars. This is shown in "Shins of the Father" (Season 1). In "Jumpin' Crack Bass" (Season 2), she does repairs on Hank's truck - and Hank is known to be protective of his truck. As the series goes on, she becomes a stereotypical "dumb" promiscuous type, with no apparent expertise in cars.
The Order of the Straight Arrow - S1-E3
Continuity mistake: When the park ranger is threatening to arrest Hank after finding them with the bird down by the small dam, the line on his hat keeps disappearing and reappearing as he talks. (00:20:20)
Hank's Got the Willies - S1-E4
Continuity mistake: When Hank is talking to Peggy in the kitchen about his guitar, she is pouring herself a mug of coffee. While the mug is on the counter, it is solid white, but when the angle changes, there's a big pink heart that appears on the mug. (00:03:40)
Hank's Got the Willies - S1-E4
Continuity mistake: In the kitchen, Hank tells Peggy he's going back to bed and asks if she's coming. She says no, and during this moment she's holding her coffee mug by the handle, but in the next shot she has her fingers through the handle, holding the mug itself. Also, the heart on the mug moves, and a red sugar container appears in her other hand. (00:03:45)
Hank's Unmentionable Problem - S1-E6
Continuity mistake: Bill tells Hank to not be embarrassed about his constipation, and Hank is holding his beer can where his finger is over the top of the label. Then in the next shot, his finger is suddenly lower with more of the label and logo showing above it. (00:04:08)
Hank's Unmentionable Problem - S1-E6
Continuity mistake: Bill and Hank are talking outside, and then Dale walks up to ask Hank if he's tried squatting. Hank opens his mouth part of the way in this shot, but then it cuts to a close-up and his mouth is closed. He turns to look at Bill, mouth still closed, then it cuts back to a wide angle and his mouth is open again. (00:04:25)
Hank's Unmentionable Problem - S1-E6
Continuity mistake: At the doctor, Hank's old teacher Mrs. Macintosh steps out of a door, and Hank is next to her. In the next close-up shot, Hank is suddenly closer to the old lady. Also, the frame around the window of the door gets larger between shots. (00:09:39)
Hank's Unmentionable Problem - S1-E6
Factual error: Hank is extremely constipated, but has a colonoscopy. He would not have been able to have a colonoscopy while extremely constipated, having not had a bowel movement in many days.
Other mistake: At Kahn's barbecue, everyone likes the burgers he cooked on his charcoal grill. Peggy is among the people who crowd around the grill. But in the episode "Hank and the Great Glass Elevator", Peggy and Bobby are shocked when they eat a burger that Luanne cooked with charcoal.
Continuity mistake: When Hank is barbecuing and turns the steak over, there are four burgers on the grill. When he tells Bill that the steak is for his father, there are only three. (00:02:50)
Continuity mistake: Bobby takes a bite of his brownie and leaves most of it on his tray when he starts to get up and the teacher comes over to talk about his attitude. A few seconds later, when he and Joseph start slamming their trays on the table, the brownie is gone. (00:17:45)
Continuity mistake: In this episode only, Hank admires his father and defends, or dismisses, most of Cotton's bad qualities. As Peggy says: "You just cannot see your father for who he is." Throughout the rest of the series, Hank and Cotton's relationship is tense and unpleasant. In the episode "The Father, The Son, and JC" (Season Six), Bobby can't even find an old photo in which Hank and Cotton are both happy.
Character mistake: In Episode 1.11, Hank says "McDonnell Douglas L-1011" when it is really Lockheed L-1011. McDonnell Douglas produced the DC-10, which was similar.
Continuity mistake: When Hank and Bobby throw toilet paper over Junie Harper's tree, Hank is only able to throw one roll over the tree and after Bobby knocks over the birdhouse they have to run away. The next morning when Hank is driving Bobby they look over to see Junie Harper trying to get the toilet paper out of the trees, but there is now a lot more than just one roll.
Other mistake: The reflecting light on the water is on the wrong animation layer, appearing over anything else in the scene that comes into their position, despite being in the background. (00:20:40)
Husky Bobby - S2-E6
Continuity mistake: Backstage at the fashion show, Andie uses his bag to knock Bobby's bag off of the makeup counter. When Andie sits down and tells Bobby to go get him a brownie, both bags have disappeared.
Answer: If you are talking about the "Thou shalt not do this, thou shalt not do that" song, I'm not sure if a song like that actually exists. Green Day, the guest voices on the episode, might have just been improvising to keep their appearance from being too obvious.