When Sheldon is eating his grilled cheese sandwich at his mother's in Texas the sandwich keeps changing from burnt to unburnt on the front between shots.
This is allegedly the first time Penny and Leonard sleep together. However, they already slept together in The Lunar Excitation.
When Sheldon and Mary have dinner, the orientation of her soda can changes between shots.
Leonard says, "You can still publish the actual results." Since it was a NSF expedition, it is not simply a decision to publish their results.
When the guys return home after being away for three months, Leonard unlocks the door, they enter, and all the lights are on. However, Penny has always had access to the apartment and may have left the lights on the last time she was in there.
When Penny is trying to comfort Sheldon, she refers to a scene of a new Star Trek movie, which would've been in cinemas while Sheldon, Leonard, Raj and Howard were on the North Pole. Penny and her other friends have never shown any interest in Star Trek, and even seem to dislike it. Therefor it makes no sense for Penny to have gone to see the movie in their absence.