Deliberate mistake: Has there ever been a more obvious body double than the one used for Angie Dickinson in the shower scene? Not only does the top of every shot stop right at the collarbones, she is much taller than Dickinson, she has a much deeper tan and (of course) her breasts are much, much bigger.
Plot hole: After his mother's death, Peter sets a camera up outside her therapist's office, so he can see who goes in and who comes out; he thinks his mother's killer might be one of the doctor's patients - a tall woman with long blonde hair. All he sees is people going in and people coming out, and then he sees her - the tall blonde. Problem: He never saw her going in, only going out, which is critical, because the killer is the therapist - a tall man - wearing a long blonde wig and make-up. Of course no-one says anything about "We never saw her going in, only out, so it must be the therapist!"
Continuity mistake: When Liz is talking to the police and Dr. Elliott's psychiatrist after Dr. Elliott is shot, she gestures many times, showing that her hands are perfectly fine. How is this possible when Dr. Elliott slashed them? Even if this conversation takes place much later, she should still have a small bandage or signs of stitches.