Factual error: The main apartment in the show is located in Pasadena, Calif. When looking out the window in the background, mountains are shown in the distance and far away. In reality, Pasadena lies right underneath the long mountain range so the mountains would be up close, not far away.
Suggested correction: Not true. I lived in Pasadena for years and in some areas, the mountains do look further away. The view from the San Pasqual apartments, you can barely see the mountains.
Revealing mistake: Most of the time, though not all the time, Leonard has no lenses in his glasses.
Suggested correction: Not obvious at any point in the show.
Factual error: Every time we see Leonard's laser lab the door is open and he is able, if the story requires it, to fire the laser. As evidenced by the several safety notices visible and the scripted requirements for characters to put on safety goggles it should not be possible to activate the laser while the lab door is open - it would be interlocked, meaning the laser wouldn't function unless the door is closed.
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.
Revealing mistake: If you look at the marble patterns on the walls, they are all exactly the same from floor to floor. Also the bricks oddities on the stairway match from floor to floor. They change lights, elevator tape etc. but it's the same floor used over and over as different floors.
Continuity mistake: In the early episodes, Leonard is depressed and buys cat stuff. He comments that he has taken in to account Sheldon's asthma with a hyper allergenic cat. Later in the series it is Leonard with asthma and Sheldon who gets all the cats.
Suggested correction: Presumably Sheldon also buys non-allergenic cats.
Factual error: Characters sometimes talk to one another on video calls, and a character (A) often watches these from an angle rather than face on, allowing the viewer to see both character A and the character on the call (B). When this happens, character B is shown turned a little to one side, to give the impression that they are looking directly at character A (as they would be if they were present in the same room). In reality, this means that character B is looking off to one side of their monitor and not at the image of character A, something that it makes no sense to do. No matter what angle the laptop is shown from, provided character B is visible, they should appear to be facing the viewer.
Suggested correction: Since we almost always see these video calls using full-screen, people sitting side-by-side would be on different sides of the screen. You wouldn't look into the webcam itself to see them. You would look at the screen, and move your head slightly to look at the left part versus the right.
Continuity mistake: In the beginning of the series Sheldon knocks on Penny's door only once. Later in the series Sheldon explains he has always knocked 3 times because he walked in on his father with another woman.
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.
Continuity mistake: Throughout the early episodes, shots of Sheldon's bathroom shows the door open and a rack of towels against the wall. Later episodes show that that area is actually where the sink and mirror are.
Revealing mistake: Any time that Leonard is "driving" his hands never move and the wheel never moves either.