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.
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'.
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.)