Deliberate mistake: Very often, during Skype or other video chat conversations, the people on the computer screen look left or right when talking to someone who is not directly in front of the screen. Although it certainly helps the viewers understand who's talking to whom (which is why I assume it's a 'deliberate mistake'), it isn't realistic, since it implies that the people on the screen are actually constantly looking away from their computer instead of at their screen or webcam.
Deliberate mistake: Whenever anyone walks across the front hallway, it takes them 6-7 seconds to walk from one stairway to the other. On the backside of the stairway, it only takes 1-2 seconds to walk across.
The Retraction Reaction - S11-E2
Deliberate mistake: Leonard is on NPR on a Friday, and the guys don't talk about it until the following Monday at work. They spend so much time together, they surely would've discussed it sooner.
The Grasshopper Experiment - S1-E8
Deliberate mistake: At the end, Penny holds up a bottle of rum to show Leonard that she added it to his drink. However, the B is missing from the name "Bacardi" (00:19:30)
Chosen answer: The song is called "Dark as a Dungeon" and was written and first performed by singer-songwriter Merle Travis in 1946. It has been performed by a wide array of artists, including Tennessee Ernie Ford, Harry Belafonte, Dolly Parton, Queens of the Stone Age, Kathy Mattea and Amy Grant. But it was made most famous when it was performed and recorded by Johnny Cash during his concert at Folsom Prison in 1968. According to Wikipedia: "It is a lament about the danger and drudgery of being a coal miner in an Appalachian shaft mine. It has become a rallying song among miners seeking improved working conditions."
Michael Albert