What Happens at Home... - S6-E10
Character mistake: After the murder that takes place while the team is at the church meeting, they gather together to talk about narrowing down the suspect list to the people who didn't show up to the meeting. Spencer shows up with a stack of papers and says 18 suspects didn't show up. We then see his actor break character and almost laughs when he looks over at Ashley, whose actress too almost breaks character from laughing.
Character mistake: While looking at a doctored photo of Lincoln and his mother, Lincoln mentions that Michael was born in 1976, but at the end of "Killing Your Number", it reads on Michael's grave stone that he was born in 1974.
Character mistake: Patterson - the oceanographer - at one point claims that Lithuania is land-locked. She should know better; in fact, Lithuania has a coastline of close to 100 km on the Baltic Sea.
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.
Character mistake: When Rich refers to the film showing Davis (decades ago), he figures they were in "Asia... Mongolia or somewhere." However, the film cannister had "Tibet, 1937" written on it - important enough to show the audience, but not for the character to read? (00:40:25)