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.
The Man in the Fallout Shelter - S1-E9
Character mistake: Hodgins is talking to Bones and says something about Goodman not being able to see his family and Zach not being able to see his kids. Goodman has kids, Zach was going to visit his family, not the other way around.
Suggested correction: With Zack's kids, Hodgins was referring to the 40 cousins and nephews that Zack has, and sees every Christmas.
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.
Flashpoint - S1-E5
Character mistake: Chief Pope threatens to take away Lt Flynn's stripes. Lieutenants don't have stripes.
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)
First Blood - S12-E9
Character mistake: Sam points out Elk Mountain, but there is no Elk Mountain in Rocky Mountain National Park.