5. Leonard Is A Dirty Liar/Hypochondriac
Sheldon might be quirky, Raj might be incapable of normal social function and Howard might be comically broken in medical terms, but none of the gang can beat Leonard for his intolerance levels. His allergies are well-known (though few are specifically explored) and it is established that he has a vast medical supply ready for any event that irritates his allergies. We do know that he is lactose intolerant and asthmatic, suffers very bad myopia, as well as suffering lesser conditions like carsickness, nosebleeds, sleep apnea, migraines, non-specific allergies and is prone to both ear infections and future heat disease. He's usually very committed to his allergies and intolerances, which is why in "The Good Guy Fluctuation" in season 5, he refuses wine because it will give him a migraine ("sulphates.") But that doesn't exactly fit with the fact that he drinks wine throughout the show, including when he and Penny are attempting to have good sex, which is presumably not the time to get a debilitating migraine.
4. Leonard Had a 27 Day Relationship That Lasted Less Than 12 Hours
At first, the show sets up Leonard's relationship with Joyce Kim as a long-term sort of thing: they had a relationship rather than a short, spectacular fling, or at least that's the implication of how they are framed in retrospect. But in the second season, when Sheldon is seeing Stephanie, Sheldon, Raj and Leonard discuss Leonard's past relationship with Kim, and it is mentioned that they went out for 27 days, before Kim defected back to North Korea. However, in a flashback sequence in "The Staircase Implementation," when Leonard is explaining the origin of his living with Sheldon, he establishes that he almost slept with Kim less than 12 hours after meeting her, and that Sheldon drove her away, before he could reveal too many secrets about his work, which presumably he could have, if that wasn't the end of their relationship entirely.
The way it's presented, it seems like that's the last time Leonard saw Joyce, which makes the whole 27 days issue extremely confusing.
3. Sheldon Can't Dance... No Wait, He's Actually An Expert
In season 4 episode "The Agreement Dissection," Sheldon ends up on girls night out, and the foursome of he, Penny, Amy and Bernadette end up going dancing. Expecting Sheldon not to be too happy with the situation, the girls question his dancing ability, which he rebuffs with a claim that he is actually a proficient dancer thanks to his Cotillion training when he was young. This might lead to a hilarious dancing sequence, but it is direct contradiction to season 1 episode "The Dumpling Paradox, in which Sheldon unequivocally states that none of the group, including himself, are able to dance.