In one of the episodes Sheldon mentions the Male Indian Peafowl as the "Male Peacock". That is a common mistake people make. The correct name for the bird is Indian Peafowl. The male is the Peacock and the female is the Peahen.
Sheldon's trademark knock (3 knocks + name, 3 knocks + name, 3 knocks + name) was not established until later in the first season, however, in a scene that pre-dates the start of the show, he is shown doing that knock sequence.
In numerous episodes, Sheldon is heard to use the phrase "off of". Example - You get OFF a bus, not OFF OF. This is grammatically incorrect, and as such, would never be used by Sheldon Lee Cooper and his eidetic memory.
There are several chronological errors in the pictorial time-line shown during the musical title sequence. For example, the Moai of Easter Island are shown as "BC" and the Giza Pyramids follow the Chichen Itza 'El Castillo'.
Penny's first car is a red 1980's Volkswagen Cabriolet Mk1 convertible. In numerous episodes it is mentioned to have the car's "check engine" light on. 1980's VWs (without computerized engine controls) had no "check engine" lights.
In Season 7, (708) everyone has Thanksgiving at the Wollowitz home (Penny learns she's married to Zach.) Raj is in the kitchen cooking as Mrs. Wollowitz is bedridden from Gout. ("Did someone say pie?") and the refrigerator doors open with hinges on the RIGHT so the camera can't film it's contents. In Season 8, (818) a power failure causes all of Howard's late mother's cooking to thaw, creating a strong emotional reaction from Howard. They decide to have everyone over for one last Mrs. Wollowitz meal. The same pale yellow refrigerator now is hinged on the left side so that the camera can film the inside of the freezer, filled with Brisket, Kugal, Meatloaf, and Chicken soup. Late that same season, (823) Howard, Raj, and Stewart are cleaning the kitchen ("It's A Hard Knock Life" Charnin/ Strouse) and the hinges are back on the right. This is an intentional goof as it's easier and cheaper to not have to expose the interior of the refrigerator; but still, we notice.
In one episode, Sheldon states that pets are banned under the roommate agreement, but in another he adopts 25 cats. But it's regularly shown that Sheldon uses the roommate agreement to his own advantage, sometimes unaware, but more often completely ignoring, how it affects Leonard.
Whenever anyone is using the stairs in the apartment building, it is obvious that each floor is the same set, slightly redecorated. This can be most easily seen in the distinctive marks on the lamps that are next to each apartment doors (especially the one on the left.)