Factual error: There is not a snowball's chance of Howard becoming an astronaut. He suffers from cardiac arrhythmia, something about which he is completely open. "Transient" or not this is easily detectable by a simple electrocardiogram, which is a fundamental part of the many medical examinations a potential astronaut will have to pass. He wouldn't even be allowed to start basic training. Note also that during his basic training he shows signs of severe stress and trauma which would inevitably trigger a cardiac episode. Even if it didn't the fact that he was falling apart at the seams would have him removed from the programme for his own good.
Factual error: Penny picks up a hand gun, pulls back the slide to ensure the chamber is empty and then releases it, leaving it closed. She then loads a magazine into the butt of the weapon and hands it to Leonard. He starts to smooch her and shoots himself in the foot. The gun could not have fired, because Penny would have to pull the slide all the way back and then release it again to load the first bullet. There couldn't have already been a bullet in the chamber, as the whole reason Penny pulled the slide part of the way back in the first place was to check just that.
Factual error: Sheldon and Amy have agreed to call each other "boyfriend" and "girlfriend." Sheldon suggests she should peruse the new relationship agreement while he sets up his notary stamp. As an agreement, both parties would sign. As a Notary, Sheldon would know that he could not notarize his own signature.
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.
Character mistake: In Amy's lab, when she is complaining to Sheldon how she gave him the simplest tasks, and that they're still not done, Sheldon replies with how Einstein failed math because he wasn't challenged, and that it is the same case with him. Einstein never failed math. Sheldon of all characters should know that.
Character mistake: The bird that Sheldon is afraid of is referred to in the scene as a "Blue Jay." In fact, the bird is a Black-throated Magpie. A Blue Jay is a little smaller, has a shorter tail, does not have the darker plumage, and can be quite obnoxiously vocal.
Character mistake: Sheldon says (during Fun With Flags) "But like the flag over Fort Sumter, I'm still here." The Sumter flag was lowered when the fort surrendered to the Confederates. The flag over Fort McHenry was never taken down despite the British bombardment and inspired Francis Scott Key to write the National Anthem about it. Whouldn't a vexologist know that?
Other mistake: When Mary is rubbing VapoRub on Sheldon's chest, she makes a comment that last time she saw that there was no hair, to which Sheldon replies "Yeah it filled in last year." In Season 1, Episode 11, Penny rubs VapoRub on Sheldon's chest and a very clear remark about chest hair was made. Season 1 for the show was 4 years ago.
Continuity mistake: Bernadette holds her wine glass in her left hand, holding it from the base of the glass; in the next shot she's drinking from her right hand holding the top of the glass and in the immediate next shot she's once more holding the glass from her left hand. (00:14:40)
Factual error: Leonard lies to Sheldon about his being over at Penny's at 3am by saying he went across the street to use the gas station bathroom. Sheldon refers to it as "The Colorado Boulevard Chevron." The address of their apartment building is 2311 N. Los Robles. Colorado Boulevard is five miles away. (Yes, it's a fantasy sequence, but even in Leonard's head, Sheldon's not likely to make a mistake like that). (00:15:45)
Continuity mistake: When Penny is giving Sheldon his haircut, she accidentally shaves the back of his head. The clippers are going from the bottom of Sheldon's hair, up to the top of his head. When Sheldon goes to leave the room, it can be seen only four or five inches from the back of his head have been shaved off.
Continuity mistake: After Howard and Bernadette are pronounced husband and wife, when they kiss the camera angle switches to an overhead shot, but as it zooms out (to a digital image) things change - such as the configuration of the rooftop, or things move - such as the cable dishes, or things even vanish - such as air-conditioning units.
Continuity mistake: In the scene where Sheldon is scooping spaghetti onto his plate, he pauses with the spoon in the serving dish. Shot cuts to Amy and the spoon is still in the serving dish. When shot cuts back to Sheldon the spoon is being held over his plate. Shot cuts back to Amy and the spoon is back in the dish.
Continuity mistake: After Amy returns to the apartment for the Grease DVD and discovers Penny doesn't like the painting, she then exits the apartment with the painting and the DVD with her. When Amy says to Penny she doesn't need her pity, the DVD is placed above the hanging string of the painting. In the next shot, as she walks into the hallway, the DVD is suddenly almost half below the string.