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)
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)
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 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.
Square-Footed Monster - S13-E3
Continuity mistake: While the neighbors are tearing down the McMansion with their tools, when Hank chainsaws the door and kicks it in, Hank's watch suddenly vanishes from his wrist mid-shot.
Bystand Me - S10-E2
Continuity mistake: When Bobby encounters Harv at the Get In Get Out mini-mart, Harv's fingerless gloves vanish and reappear a few times until Bobby leaps into Dale's van. (00:11:00)
Bystand Me - S10-E2
Continuity mistake: While Bobby's on the sign-up line, Dale complains it's not fair that only kids get paper routes, and Dale's watch vanishes from his left wrist, then reappears. The watch style also changes in the shots facing Dale. (00:03:20)
Continuity mistake: When Dooley is saying "She still wants to dance," Clark is standing next to him and they are roughly the same height. When the shot changes, Clark is suddenly a few inches shorter than Dooley.
Bills are Made to be Broken - S4-E3
Continuity mistake: When Hank and Bill are sitting on the hood of Bill's car and talking, one shot of the football field shows it to only be about 70 yards long between the end zones.
Three Coaches & a Bobby - S3-E12
Continuity mistake: When Hank sees Coach Sauer driving his car on the field, you can see Sauer's car has a red front bumper. When we see the front bumper again a few shots later it is grey. (00:15:30)
Three Coaches & a Bobby - S3-E12
Continuity mistake: If you look carefully during Coach Sauer's training run while Bobby is running down the stands, you will quickly see the mud disappear on Bobby's face and reappear in the next frame. (00:08:35)
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.