Character mistake: Hank seems surprised to learn a lot of the Alamo's defenders weren't from Texas. Given that kids study this in American history, especially Davy Crockett's role in the historic event, it seems unlikely he'd be ignorant about this.
Bystand Me - S10-E2
Continuity mistake: While Hank, Peggy, and Bobby are asking for help picking up the papers with the dangerous ammonia and bleach recipe, a person in back of the group they are talking to that was originally wearing a dark red hat, red shirt, and blond hair switches to a pink, light blue shirt, and brown hair. A window also suddenly appears behind this individual, where there wasn't one before. (00:17:49 - 00:18:11)
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: Hank talks about watching the Broncos vs the Cowboys in Super Bowl XIII. This is incorrect. They played in Super Bowl XII. Super Bowl XIII was the Cowboys vs the Steelers.
Megalo Dale - S7-E10
Continuity mistake: Chuck Mangione is shown living in the Mega Lo Mart, having faked his death after the explosion. Yet, he was shown in "Death Of A Propane Salesman" at Buckley's funeral.
Square-Footed Monster - S13-E3
Other mistake: When Lucky drives off in the pickup with the tow chain, as the McMansion crumbles to the ground, Luanne runs away from the house, and part of her legs are missing. The animation is incomplete.
Other mistake: When Hank tells John Redcorn that his family is not Jewish, John is noticeably surprised. However, in the episode "Revenge of the Lutefisk" (3x21), John was at church with the Hills and Gribbles. He already knew that they are not Jewish.
Continuity mistake: The entire episode is about Hank being very attached to his current truck, and refusing to purchase a new one (until he has no choice). But in S4 E23 Hank tells Peggy "You know how much I want a new truck."
Death of a Propane Salesman (2) - S3-E1
Continuity mistake: When Peg is on the phone with Buck the wall telephone has a long cord, then she hangs up the receiver on the wall, but in the following shot both the receiver and its long cord are weirdly missing from the wall phone. (00:05:15)
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)
The Year of Washing Dangerously - S10-E9
Continuity mistake: Peggy is looking out the kitchen window into the Gribble household, which is wrong. Khan lives on the kitchen side of the Hill household. The Gribbles live on the garage side.
Nine Pretty Darn Angry Men - S3-E7
Continuity mistake: When Peggy's shoe sole comes off, it comes off the right shoe, but in the repair shop she hands her left shoe to the cobbler.
Character mistake: Hank says that his parents haven't spoken in four years. However, Cotton and Tilly were both at the Hill home in the episode "Nine Pretty Darn Angry Men" (Season 3). Cotton bullied Tilly on that occasion. And four years has not passed in between those episodes, or else Bobby would be a high school student.
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.
Born Again on the Fourth of July - S13-E14
Factual error: When Bobby and Lucky first walk into the church service that is taking place inside the karate studio, the pastor is calling everyone a sinner, and proceeds to point to a picture of Bruce Lee behind him and says he was a big sinner who repented and knew that the unrighteous would not inherit the kingdom of Heaven. Bruce Lee was an atheist.
Death Buys a Timeshare - S9-E3
Plot hole: During a heatwave, the Hills are dying from the heat and even look for a pool to swim in to cool off.This makes no sense, they live in a ranch house which would have central air or at worst some window air conditioners.
Serves Me Right for Giving General George S. Patton the Bathroom Key - S13-E15
Other mistake: Here, it's made into a big deal that Cotton wants to be flushed in the same bathroom Patton used. Yet in Season 4, Episode 2, "Cotton's Plot," he successfully gets a cemetery plot in a Texas military cemetery.
Square-Footed Monster - S13-E3
Continuity mistake: While Hank is at Arlen City Hall, asking about forms for a legal petition, the first clerk tells him that Hornsby has been dead for 11 years. The filled directory board is visible on the wall. However, when the second clerk says Hornsby has been dead for 18 years, the directory board is entirely blank.
Square-Footed Monster - S13-E3
Continuity mistake: When the four "day laborers" hop into Eric's pickup, then stop at Mrs. Dwyer's house, there is no doorknob on the door in the wide shot. However, a doorknob appears in the next closeup.
Square-Footed Monster - S13-E3
Continuity mistake: When the McMansion starts breaking apart, endangering the other homes, Dale's watch is on his left wrist, as usual. But when the neighbors arm themselves with tools, Dale's watch is on his right wrist (it's not a flipped shot), then it returns to his left in the next shot.
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.