Continuity mistake: When Hank's trying to convince his dad to let the Vietnam vets join the VFW, the table behind him with the bowl of fruit and the vase of flowers on it disappears between shots. (00:12:45 - 00:13:15)
Other mistake: At the prison, Bill meets Alicia, the same woman who gave Leanne a ride to the Hill house in Season Two ("Leanne's Saga"). Same voice, tattoo on the arm, and general appearance. Alicia tells him that she was in prison for killing her boyfriend. So it's odd that she was free in Season Two. Most likely, a sentence for homicide or manslaughter would be at least five years, and that much time has not passed. Otherwise, Bobby, Connie, and Joseph would be in their late teens.
Returning Japanese (1) - S6-E21
Plot hole: In this episode, Hank meets his paternal half brother. They look alike. However, we saw in an earlier episode that Hank's facial structure takes after his mother, not his father. Since they are paternal half brothers, then the Japanese brother cannot look like Hank since he has a different mother.
Returning Japanese (1) - S6-E21
Factual error: As the Hills get off the plane in Japan the camera crew is waiting for them as they exit the terminal however they do not go through any sort of customs area. Every international flight you must go through customs before entering the country.
Returning Japanese (1) - S6-E21
Continuity mistake: On the plane whilst Bobby is watching the TV in front of him, you can see a man to the left of Bobby and an empty seat to Bobby's right. The seat arrangement looks to be like the middle row of seats where there is 3 seats instead of two. After the shot shows footage for the "Haiku" movie the shot changes to Bobby and the man sitting next to him but there is now no empty seat next to Bobby. It appears that he is now sitting at a window seat.
Returning Japanese (1) - S6-E21
Factual error: During the scenes where the Hills arrive at the airport in Japan, there is no flight crew outside or around the terminal when they exit. Whilst they may be the last people to exit the plane, there will be one or more crew members to farewell passengers and also to close the terminal door. There doesn't seem to be any crew anywhere close, which seems particularly odd especially as Hank and Cotton manage to just stroll back onto the plane. A little after they leave the camera crew just go right through the terminal onto the plane without anyone there to stop them.
Returning Japanese (1) - S6-E21
Factual error: In the airport scene in Japan, the news crew arrives to film Cotton Hill. As Cotton Hill gets back on the plane trying to escape the news crew, we see a shot from outside the plane and the news crew is out on the tarmac. With security regulations at airports a film crew wouldn't be allowed to just go out onto the tarmac.
Continuity mistake: In the scene where Bobby is in the closet, he takes off his shoe.When Hank walks in and pulls him out, Bobby's shoe is back on.
The Fat and the Furious - S7-E2
Continuity mistake: When the woman at the hot dog contest says Bill is a freak and he gets depressed, he spits out the dog. On his plate there only a few hot dogs. After Hank talks to Bill, there are many more.
Bad Girls, Bad Girls Whatcha Gonna Do - S7-E3
Other mistake: When Bobby is caught spray-painting the fence, Hank says that he and Peggy took their wedding photos in front of that fence. In the episode "As Old As The Hills", part one, there is a photo of Hank and Peggy living in an apartment after they got married.
Continuity mistake: After Hank calls Bobby into the room to talk about the woman's magazine Peggy discovered, you can see Peggy's hair is a little messed but nothing in it. A few shots after Hank asks Bobby about making the turkey there is a close up shot of Peggy and she has a few stains in her hair which disappear in the next shot. The stains look very similar to the ones she gets a little later on when she falls on the turkey.
Continuity mistake: The door to the slaughterhouse (which Luanne blocks with the barrel) changes configurations between shots. When Trip is trying to get in, the exterior is blue with a doorknob. After Peggy busts in, it is white with a pushbar across the middle. The interior is white with a pushbar in all shots.
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.
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.
I Never Promised You an Organic Garden - S7-E18
Continuity mistake: During the scene where Peggy still finds insects over the tomatoes, she goes to show Hank one of the tomatoes. As she is showing the tomato it has around six or seven flies buzzing around it. The next shot shows Peggy from behind still holding the tomato, and you can see there is not one fly around the tomato anymore.
Deliberate mistake: Hank only discovers the tattoo on the back of his head after having his hair shaved off. There is no way, however, he wouldn't have noticed it before this. The tattoo artist had to shave hair off the back of his head before putting on the tattoo. While he might have not realised right away, the pain the following day and the fact that he would have noticed a large part of his hair missing would make it unavoidable.
Plot hole: Amazing that neither Hank's mother nor father ever saw the bandage on the back of his head, peeled it off, and would discover his tattoo.
Other mistake: Dale laughs at Nancy for picking up the diazinon, instead of the bird repeller gel. He then picks up a bag, goes up the ladder, and when he turns the bag around, it says diazanon on the bag he picked up. The only gel in the scene was what Nancy picked up.
Continuity mistake: Peggy's shoes go from purple to white in one shot in Nancy's kitchen.
Continuity mistake: At the end when Laoma is crying and hugging Bill, the tears on her face are on her cheeks. As she asks, "Wait, Astronaut who?" the tears are now just found near her eyes. As Bill explains, she has her hands on his shoulders. She then turns and the tears are not even in her eyes now.
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.