Continuity mistake: It is stated multiple times that forensics reports showed that the Bay Harbor Butcher used ketamine to incapacitate his victims. However, it wasn't ketamine that Dexter used in the original series, but rather M99.
Continuity mistake: When Miranda enters the bar the cleaner appears/disappears randomly between shots.
Factual error: Season 2, Episode 1: Cleon is fighting some "ninja" warriors. One of them slices part of an android's head clean off, below the right eye. When the android is next seen - with part of its head cut off - the slice is above the right eye. The right eye is fine.
Factual error: When Alina stands outside of the door to the banquet hall, waiting to demonstrate her power to the king and queen, there is a modern electronic occupancy sensor on the wall behind her.
Factual error: A passerby tells Sheldon Sampson that the stock market has just gone "tits up." This is a British expression that you would likely not hear in the United States, and you certainly wouldn't hear it in the 1920's.
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.
Other mistake: Vincent falls from a parking garage and smashes onto the roof of his car, appearing nearly dead. In the next episode, he is in the hospital with multiple broken ribs and other fractures, yet he is released without even staying the night for observation.