The Long Distance Dissonance - S10-E24
Continuity mistake: When Sheldon takes a plane to see Amy in Princeton, the plane he is on has 2 engines when it takes off and 4 engines when it lands. It's a direct flight from LAX so he wouldn't have changed planes.
The Thespian Catalyst - S4-E14
Revealing mistake: When Sheldon goes to Penny's apartment and they do a scene study together, they both read lines from a multi-page script that Sheldon brings with him. On a couple of occasions the camera cuts to a side view of Sheldon and we get to see the page of the script that he's currently reading from. The pages are completely blank.
The Nerdvana Annihilation - S1-E14
Factual error: The time machine in this episode is supposedly the same one used in the H.G. Wells movie, but there are mistakes in the controls that would have been obvious to the characters: 1. When Leonard first gets into the machine he talks about traveling into the past and says something like "I'm setting the date to..." That is the method used to operate the Wayback Machine in the Rocky and Bullwinkle cartoon, but not the time machine in the H.G. Wells movie. Rod Taylor could not set the desired date on the controls, he simply used the lever to move forward and backward through time and the date display updated automatically to show the current date. 2. In the movie the default position of the lever was at a 45 degree angle. Pushing the lever forward moved the machine into the future, pulling back on the lever moved the machine into the past. In the TV show the default position of the lever is vertical and Leonard pulls the lever backwards to move forward into the future - opposite of the movie.
The Spoiler Alert Segmentation - S6-E15
Revealing mistake: Throughout the series much is made of the fact that Howard's mother is grossly obese, and in The Engagement Reaction (S4 E23) it is even mentioned that her arms are hugely oversized. However, at the end of this episode we see her arms when she grabs Raj to stop him climbing out of the window, and they are the arms of an average-weight woman - they aren't overweight at all.
Continuity mistake: Penny throws a black iPod out the window, but when Raj comes in having found it on the ground outside he has a silver one. (00:01:00 - 00:02:45)
Other mistake: Sheldon would not have been able to crawl straight along the window ledge to Leonard's bedroom due to the layout of their apartment. Leonard's window is round a corner from Sheldon's.
The Agreement Dissection - S4-E21
Continuity mistake: When the girls take Sheldon out dancing, he mentions that he considers himself a proficient dancer due to his Cotillion training when he was a boy. But in S1E7 The Dumpling Paradox, when Leonard suggests that they could have gone dancing with the girls, Sheldon said that nobody in the group - including him - could dance.
The Pancake Batter Anomaly - S1-E11
Factual error: Leonard's myopia is greatly exaggerated in this episode. Even with high myopia he should still be able to navigate his apartment without aid especially after living there for seven years. (00:17:40)
The Nerdvana Annihilation - S1-E14
Character mistake: Being the first to play in the "time machine," Leonard says he will set the date for March 10, 1876. Howard says "Good choice. Alexander Graham Bell invents the telephone and calls out for Doctor Watson." Bell's assistant, Thomas A. Watson, was an experienced electrical designer and mechanic, but he did not hold an advanced degree. It is well documented that the first words transmitted over the telephone identified him as Mister Watson.
Revealing mistake: In the scene where Leonard is supposedly playing Halo 3 against somebody in Copenhagen, there is no ethernet cable or wireless adapter connected to the Xbox 360 (one of the originals without wireless inbuilt), which means that there is no way he can be playing online with someone in another country. (00:09:00)
The Platonic Permutation - S9-E9
Continuity mistake: In episode 16 of season 1, Penny is talking about making a cake for Leonard's birthday and states she knows his birthday because she was checking a horoscope for him. But in this episode she does not know when his birthday is, which leads to Leonard revealing he was secretly reading Penny's diary. (00:40:00)
Suggested correction: It's certainly possible to forget something you knew eight years ago.
This is perhaps true for two people who rarely have contact, but I would list this as a character mistake. Penny and Leonard are in a serious relationship and are married at this point.
This is also a couple who BOTH forgot it was their anniversary.
I forgot my wife's birthday on a number of occasions. It's one of the reasons she is now my ex-wife.
There's a difference between forgetting and not knowing.
The Cushion Saturation - S2-E16
Factual error: Leslie Winkle takes shelter with Howard in the middle of a paintball game. Her gun has no hopper for paintballs and thus no ammo, so it's useless, despite her apparently using it outside seconds earlier. This isn't a model with an internal magazine - the port to attach the missing hopper is plainly visible.
Continuity mistake: Sheldon puts the bag in the bin at the start of the episode so that the handles are facing the ground. When he gets the bag out, the handles are at the top of the bin.
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.
The Vartabedian Conundrum - S2-E10
Continuity mistake: When Leonard and Penny are talking in the dryer room, the size of the towel Penny is folding keeps changing. (00:15:00)
The Shiny Trinket Maneuver - S5-E12
Continuity mistake: Howard, Leonard and Raj play Jenga in a strange sort of way. Most notable is the playing order, 1. Howard, 2. Raj, 3. Raj, 4. Leonard, 5. Howard. When it's Leonard's turn he doesn't move any bricks, only pokes the tower and it's then Howard's turn. When its Raj's first turn he puts a brick on the top of the tower. When it's his turn again, the brick is gone and he puts a new one in its place.
The Retraction Reaction - S11-E2
Character mistake: When speaking on the radio, Leonard says he has never heard himself through headphones. We have seen him several time wearing headphones while speaking, for example during a podcast with Will Wheaten and on Science Friday too.
The Big Bran Hypothesis - S1-E2
Continuity mistake: At the end of the episode, when the gang is reading the furniture assembly instructions, a pencil in the cup in front of them suddenly vanishes. (00:19:00)