Character mistake: When Beth and her mother discuss the agent's fee of 15%, Beth says it is $49.54. However the prize money was $500, the expenses were $172.30, which leaves them $327.70. 15% of $327.70 equals $49.16. (00:10:00)
Character mistake: Dr. Agnes Jurati, one of the Federation leading scientists, is bored during the hyperspace travel, and so she chats a bit with the captain. While she gives her quirky speech, she casually mentions that "there are over 3 billion stars in our galaxy." She's not wrong, technically, but the number of stars in our galaxy is estimated between 100 and 400 billion. She is way off. (00:08:40)
Suggested correction: Two things. You state that she's not wrong, which she isn't. The fact that she chose an odd turn of phrase doesn't make it a mistake. Plus, you reference the number of 'galaxies' in our galaxy but I am guessing this is just a typo.
Character mistake: The FBI team responds to a murder crime scene. The victim is lying face down in the bed with blood everywhere. The team leader takes the victim's wrist with his bare ungloved hand. A trained FBI agent would never touch a murder victim without gloves - one to protect himself and two not to contaminate the crime scene. (00:10:00)
Character mistake: Owen Strand refers to himself as "Austin PD" instead of "Austin FD" while attempting to rescue people from an armored truck. The character is a fire captain. (00:09:14)
Angel of Mercy: Part Two - S1-E4
Character mistake: When Lenore Osgood is speaking on the phone with Charles Wainwright about killing Dr. Hanover, she mispronounces "Larynx" as "Lar-nix."
Character mistake: On the classroom blackboard, "government" is misspelled as "goverment." (00:08:00)
On a Wing and a Prayer - S4-E7
Character mistake: FO Woodham drops James in Darrowby. James gives a proper RAF salute (palm forward). Woodham responds with a US style salute (palm down).