1. In "The Dumpling Paradox" (Season 1 Episode 7), the boys start Halo night 6 minutes late, and determine if the 6 minutes should be taken out of game time, bathroom time, or pizza break. Wednesday is Halo night, so it must have been Wednesday. But pizza night is on Thursday.
2. In "The Bozeman Reaction" (Season 3 Episode 13), Sheldon allows Howard to install a giant net into his apartment in order to electrocute robbers. But, In "The Ornithophobia Diffusion" (Season 5 Episode 9), Sheldon reveals that he has a fear of nets. Since "The Ornithophobia Diffusion" is after "The Bozeman Reaction", Sheldon's fear of nets may have been triggered because of being trapped in a net at the end of the other episode.
3. In "The Maternal Capacitance" (Season 2 Episode 15), Penny revealed through her conversation with Leonard's mother at the end that her father's name is Bob. However, in "The Boyfriend Complexity" (Season 4 Episode 9), when her father visits her, he introduces himself to Leonard as Wyatt.
5. Sheldon says that Amy doesn't like to wear costumes in "The Justice League Recombination" (Season 4 Episode 11). However, she does dress as a pretzel, a Star Fleet medical officer, Snow White and more in various episodes. She wore the Star Fleet medical officer outfit for Sheldon to like her more and the pretzel as a part of Sheldon's web show "Fun with Flags". The greatest counterexample is in "The Holographic Excitation" (Season 6 Episode 7) when she asks Sheldon to join her in a couple's costume pair.
6. In "The Thanksgiving Decoupling" (Season 7 Episode 9) and "The Proton Transmogrification (Season 7 Episode 22), Sheldon says that his father died when he was 14 years old. However, in "The Pancake Batter Anomaly" (Season 1 Episode 11), Sheldon states that he was once sick in Germany when he was 15 years old, and his mother wasn't able to fly to take care of him because she needed to help his father as the house had been damaged in a tornado. Also, Sheldon mentions that his father tried to give him a beer for his high school graduation, but he was 11 and his mom said no. This creates a mistake as Sheldon never attended high school and instead started college right after leaving elementary school.
7. The famous flashback episode, "The Staircase Implementation" (Season 3 Episode 22), details Leonard moving into Sheldon's apartment seven years prior to the episode, putting the events of the flashback in May 2003. However, in "The Stag Convergence" (Season 5 Episode 22), when Sheldon is comforting Howard, he says, "Also, there was a single cube of chicken bouillon in the cupboard when I moved in and it’s been bothering me for the last eight years. So, as they say, two birds." The Stag Convergence took place in May 2012, making Sheldon's reference of him moving into the apartment eight years ago at 2004, a year after Leonard is supposed to have moved in. Theoretically, one could claim the cube of chicken bouillon did not bother Sheldon for a full year after he moved in but that seems unlikely for our lovable Sheldon Cooper.
Note: Checkout the Goofs sections for TBBT episodes on the Internet Movie Database and the Contradictions page in this wikia.