Continuity mistake: When Sam is in the bathroom holding up two sample paint chips, in the first shot she's holding up the yellow in her left hand and the blue in her right hand, but in the next shot they've switched and are in the opposite hands, just as Jay arrives with the coffee.
Factual error: Pete died in 1984. Yet mentions the movie 'Ghost' with Patrick Swayze which released in 1990.
Suggested correction: It is established that the ghosts have watched television, listened to the radio, smelled pizza/pepperoni, etc. It's not impossible that Pete saw Ghost on TV.
While they do watch television now, it's unlikely that Sophie watched "Ghost" and that Pete saw it alone. The reply from others was "you know we don't know what that is" and he said "oh, right the movie thing." Which was a call back to earlier when Trevor was trying to explain who Tara Reid was and the older ghosts had a hard time knowing or remembering what a movie was.
Farnsby & B - S1-E18
Other mistake: The ghost in the Farnsby's house knew Ronald Reagan as an actor but in 70 years she didn't know he was elected President even though she would have heard about it from people who lived in the house during his terms.
Continuity mistake: At the foot of the stairs, Higgentoot is facing away from the group, and his hands are by his sides. As he turns, they're suddenly behind his back.
Factual error: The ghost doctor says he's been in that building for 85 years, meaning he would have died in 1936, but the stethoscope he wears was developed in the 1970's.
Dinner Party - S1-E4
Revealing mistake: Issac is reflected in the cabinet door when Sam is asking for the ghost's help.
Suggested correction: When was it established that the ghosts' reflections aren't seen?
Attic Girl - S1-E17
Other mistake: When Stephanie shows her chainsaw wound to Sam, it is bloody and grotesque but when Flower, who was mauled by a bear, showed Sam her wounds, they were mere scratches with no trace of blood.