Continuity mistake: When Dave and Vince arrive at the scene of the first Trouble, Dave is talking to Duke, and the camera changes from in front of Dave to just behind him. Dave wears glasses and in the front shot he has them on, but in the rear shot, they're missing.
Factual error: The 1725 issue of 'The Haven Herald' used as part of every episode's title sequence is packed with errors. It is written in modern English, with some very modern turns of phrase (e.g. Had it with this town, get on with her life). The same paragraph appears twice. It is headed "Haven, Me." - Maine was part of Massachusetts until 1820 and the abbreviation was not commonly used at all until the 20th century. And it costs 2 cents - until 1793 British-style pounds, shillings and pence were used throughout America. Also, February 18th 1725 was a Sunday, not a Tuesday.
Continuity mistake: When the taxi stops to pick up the woman, it stops with its hood in front of the door of the building beside it. She gets in and finds the driver dead, and jumps out. When the police are checking it, it has moved a full car length forward. The driver put it in park when he stopped, so it didn't roll forward.
Other mistake: Duke is checking old copies of the Haven Herald around the time of Lucy (27 years ago). A headline on the one he's reading is about Lucy's disappearance. A moment later, he's reading a story about a meteor shower that occurs every 27 years; but the text we can see states that it just happened, and refers to dates in 2010 (having just happened, not '27 years from now'). This paper is supposed to be a 1983 edition. Also there is fake Latin 'filler text' printed in some columns.
Factual error: When Audrey turns on the car radio to listen to the baseball game score, the LCD display turns to the time. When she turns it off, it shows the tuner. This is the opposite of how car radios function - the sound editor must have been dubbing in the sound, and missed this).