The Russian Rocket Reaction - S5-E5
Character mistake: Sheldon says he drove for hours in a bus to meet Wil Wheaton. He meant he rode on a bus. Sheldon never would've made that semantic error.
Character mistake: Raj cannot talk with a woman around, but when Leonard claims they'd have to cut him in half to play teams, he blatantly argues back, and Penny is present with them. Regardless of if he was facing her or not, he shouldn't have said anything.
The Clean Room Infiltration - S8-E11
Other mistake: When Howard, Leonard, and Raj are arguing in the clean room, Howard says that Leonard is jealous because Howard and Leonard used to be "best friends until Koothrapali came along." Yet, in the earlier seasons, we know that Howard and Raj already knew each other when Leonard met them for the first time and brought them back to the apartment to play video games. Meaning they couldn't have been best friends before Raj came along.
The Werewolf Transformation - S5-E18
Character mistake: In S5-15, Sheldon is asking his friends to take him to SuperCuts. But in this episode, he goes to Mr. D'Onofrio's barber shop where he says he's gone to since he moved to CA, and he said D'Onofrio had his "haircut records."
The Hamburger Postulate - S1-E5
Continuity mistake: Leonard and Leslie are rehearsing the violin and cello in the guys' living room, and when they leave the room to go to Leonard's room, Leonard's cello and stand are still in front of the comfy chair. When Sheldon gets home, the cello and stand are gone, and only Leslie's closed violin case is left on the sofa. And she picks it up and leaves with it. How did the violin get back in the case.
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 Retraction Reaction - S11-E2
Character mistake: When Leonard arrives at the Human Resources office, he tells the HR person that he's not used to being called into the Dean's office. He's not in the Dean's office, he's in HR.
The Convergence Convergence - S9-E24
Other mistake: Howard believes he's being followed by the government, when it's actually Leonard, Penny, and Beverly in their car. The shots from Howard's car show only Leonard's silhouette. The other two do not appear as silhouettes.
The Higgs Boson Observation - S6-E3
Factual error: Sheldon tells Alex that he's run a background check, including her medical records. There is no way he could obtain medical records without a HIPAA authorization signed by Alex herself. It's illegal.
The Countdown Reflection - S5-E24
Continuity mistake: Penny mentions that she knows lots of people who got married in Las Vegas. In episode 7/9, she discovers that her "fake" wedding to Zach in Vegas was real.
Continuity mistake: Sheldon puts the bag of takeout into the lobby trash can and as he does so, the whole bag turns sideways. At the end of the scene, he takes the bag out of the can and it's upright without him turning it.
Continuity mistake: When Leonard is consoling Penny on the couch, the blue Afghan she has with her constantly changes position, including jumping from the floor back to the couch.
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.
The Countdown Reflection - S5-E24
Continuity mistake: Near the beginning of the episode, Howard said the launch was on the Friday before his planned wedding. They decide to marry before he launches, and they say they only have two days. Then he tells the clerk at City Hall that he launches on that Sunday. Too many inconsistencies.
The Countdown Reflection - S5-E24
Other mistake: Raj doesn't get over his selective mutism until the end of season 6. However, he very prominently speaks during Howard and Bernadette's wedding without alcohol or meds. In the season 6 episode where he finally does speak without any assistance, Penny makes a big deal about it so it's shocking when he does it here with no big shocks.
The Friendship Contraction - S5-E15
Character mistake: Sheldon told Leonard at lunch that they were no longer friends, only roommates but later that day, he told start moved up to ninth place on his friends list because he dropped someone (can't remember who) off his friends list). In actuality, it would've bumped him up to eighth place because he kicked Leonard off the list.
The Launch Acceleration - S5-E23
Factual error: In all the episodes leading up to Howard and Bernadette's wedding, Howard was going into space to install a space toilet. But in this episode, Howard tells her they want his telescope, so he has to go.
Suggested correction: The space toilet was something Howard designed that NASA installed back in season 2. His mission to the ISS in season 5 was always about his telescope. In episode 5, "The Russian Rocket Reaction", Howard says NASA picked his team's design for the deep field space telescope that's going on the International Space Station in the spring.
The Proton Displacement - S7-E7
Character mistake: When Raj shows Penny, Amy, and Bernadette how to make jewelry, Penny keeps having trouble with the glue and gets it all over her fingers. However, in Season 2, episode 5, she has no trouble at all with the glue when making 1000 Penny Blossoms.
The Speckerman Recurrence - S5-E11
Character mistake: Throughout the show, Sheldon denounced his mother's religion and God. To the extreme. In this episode, he said, "More sense than God gave a mouse." He would never invoke God for any reason.
The Big Bran Hypothesis - S1-E2
Character mistake: Sheldon does not do his 3x OCD knock on Penny's door.
Suggested correction: That quirk wasn't brought into the series until halfway through the first season. The one-knock Sheldon happened a couple of times before it became the three knock habit.