God Is in the Details - S2-E10
Character mistake: When Diane quotes the scripture, she says "Revelations." However, it's just "Revelation." Someone as smart as her who has read the New Testament (or heard it quoted) would know the correct word.
Character mistake: In the epilogue, Charlie illustrates the relationship of mathematics to real life by discussing the golden ratio and its presence in a host of real-life situations, including the shape of a nautilus shell and the distribution of petals in a flower. These claims are disputed by some mathematicians (see http://en.wikipedia.org/wiki/Golden_ratio#Nature), and the one about the nautilus shell one is untrue (see http://en.wikipedia.org/wiki/Golden_spiral#Spirals_in_nature). It's possible Charlie is just repeating what he has read or heard without investigating it for himself, but this is uncharacteristic of him.
Smoking Gun - S1-E2
Character mistake: When the bad guys put the generator in front of the air vent, Nigel looks at the white smoke and pointing at it says "Carbon monoxide." Carbon monoxide is colorless, white smoke is mostly condensed vapor. (00:35:20)
Character mistake: Upon finding the Spanish flu virus missing, Hardison freaks out about it killing 50m people 100 years ago, saying it would be even more deadly today due to the higher population. There's no reason to think so - the death toll in 1918 was massively exacerbated by the aftermath of WWI. Not to mention today we have vaccines, plus most deaths then were actually caused by bacterial pneumonia, which today is easily treated with antibiotics.
Devil Music - S5-E4
Character mistake: When they pulled the bullet out of the wall of the Five and Dime, Danny says that it was either a .40 or .45 caliber. The .40 caliber was not produced until 1990, 37 years after the date the crime took place. (00:30:00 - 00:35:00)
Character mistake: Nick asks Janette if she's ever turned anyone else into a vampire - an odd question, considering the fact that he already knows she has. He was there when she brought the baroness across in the 18th Century flashback sequence for "If Looks Could Kill." It's even odder that Janette tells him she's never done it, since she knows that Nick knows she has. Apparently, the only one not in the know here was the scriptwriter.
The Case of the Sun Bather's Diary - S1-E17
Character mistake: The reporter over the radio says Arlene Dowling is accused of murdering Frank Ballard, but the name of the murdered guy is George Ballard. Frank is her imprisoned father, Frank Dowling. (00:43:50)
Mine Your Own Business - S1-E4
Character mistake: When Shaggy and Scooby order Sarsaparilla at the saloon, Scooby says "Two" but he appears to be holding up three fingers.
Character mistake: Nathan suggests to Dwight and Audrey that maybe he should shut down the meeting of troubled people. As chief, he should know he has no authority to stop citizens from assembling.
Character mistake: The voice tells Tom that the alarm deactivation code is 47223. However, when Tom enters the code subsequently to deactivate the alarm, he only enters "4723." You can tell he doesn't enter a fifth digit because the keypad chimes each time a digit is entered.
Character mistake: Mahershala said the child found in the cement had "osteoportrosis", instead of osteoporosis. Then the grief counselor repeated the same incorrect word.
Suggested correction: They said "osteopetrosis," which is a different medical condition from osteoporosis. Osteopetrosis is a genetic condition that causes overly dense, brittle, and misshapen bones starting in infancy.
Episode #1.3 - S1-E3
Character mistake: Richie Gulliver is referred to as Rory Gulliver on the newspaper headline. As he's supposedly a very well-known Member of the Scottish Parliament, it's unlikely a newspaper would make a mistake like this.