Factual error: During the final scene of this episode, Sheldon steps out of his office wearing a gas mask and engages Leonard in conversation where he then states he is making hydrogen sulphide gas (more commonly known as H2S). Leonard correctly identifies this as highly flammable. However, this gas is far more dangerous and is harmful to the human body at as low a concentration as 20 parts per million (ppm), can cause permanent damage at 100ppm and is fatal after two breaths at 500ppm. So for Raj (and his bird) to be completely unharmed in a high concentration of H2S (noted by the fire) is impossible.
Character mistake: The proposition that danishes wouldn't exist if Copenhagen were flooded is wrong, because "danishes" were created by Austrian bakers. They are called danish in English-speaking countries because they're very popular in Denmark. The Danish word for this pastry is wienerbrød, which translates to "Viennese bread."
Revealing mistake: When Sheldon goes to Penny's apartment and they do a scene study together, they both read lines from a multi-page script that Sheldon brings with him. On a couple of occasions the camera cuts to a side view of Sheldon and we get to see the page of the script that he's currently reading from. The pages are completely blank.
Continuity mistake: When the girls take Sheldon out dancing, he mentions that he considers himself a proficient dancer due to his Cotillion training when he was a boy. But in S1E7 The Dumpling Paradox, when Leonard suggests that they could have gone dancing with the girls, Sheldon said that nobody in the group - including him - could dance.
Other mistake: When Sheldon and Leonard are playing Archery on Wii Sports Resort, Sheldon presses the Z button on the nunchuck controller to fire the arrow, which is incorrect. When Leonard plays, he holds the Z button as he prepares to fire, then releases the Z button to fire, which is the correct controls.
Continuity mistake: Near the end of the episode, in the canteen scene, Howard shows the card trick to Sheldon again. The card at the bottom of the deck is the 2 of hearts as he takes them from the box. Sheldon then takes this card from near the top of the deck when Howard puts them on the table. The deck is not shuffled between times.
Revealing mistake: Sheldon's laptop has gone through several strange transformations throughout the series, but in this episode it takes the biscuit - it appears to be a Dell XPS with the back covered up, but the shot of Sheldon chatting only to Amy shows a Mac keyboard, with the laptop running OS X.
Character mistake: As they drive to Todd Zarnecki's to confront him, Sheldon says, "I almost feel sorry for the poor fool, sitting in his split-level suburban ranch..." Architecturally, split-level (technically three levels joined by half flights of stairs) and ranch (single level) are separate and distinct designs. There is no "fusion" of the two. I can't help but think Sheldon would have corrected anyone who said the same thing.
Continuity mistake: When Sheldon and Raj are arguing about having a desk in Sheldon's office, Sheldon's bottle of water keeps rotating without anyone touching it - take note of the label. The level of water in Sheldon's drink also keeps changing between shots.
Character mistake: Even with Penny and Zack the guys are one man short to be the original JLA because they they don't have the Martian Manhunter. If they used the same costumes in the previous year, they would have been three people short of the original line-up.
Character mistake: When Penny says she smells Chinese food and Sheldon corrects her by telling her she's slipping because it's Thai, Sheldon is eating the food with chopsticks, even though in a previous episode he uses a fork and spoon because that's how Thai people eat their food.