A New Day in the Old Town - S2-E1
Other mistake: The killer from another dimension shape shifts into the appearance of the victim, leaving the victim's body. But it transforms into Charlie's appearance and leaves behind its former female nurse shape, hiding Charlie's body to dispose of later. It would have needed to transform from the nurse shape, not out of nothing. The original nurse's body was not at that location.
Suggested correction: The shapeshifter transformed into Charlie using his dead body which it then disposes of at the end of the episode.
But when the shapeshifter transforms into Charlie, he shouldn't be leaving the body of the nurse behind since he had killed the nurse before entering the hospital and transformed there into her. The only dead body in that last scene should be Charlie's.
A New Day in the Old Town - S2-E1
Other mistake: Just after Olivia wakes up in hospital, she's trying to explain what happened, and we hear a heart monitor getting faster and faster - the doctor looks concerned and says they need to get her pulse down. But when we cut to the monitor itself it shows her heart rate as 72 - a perfectly normal value.