Continuity mistake: The bullet hole in Davidson's chest is not at all where he was shot.
Factual error: When the 747-aircraft of the fake German airline is being shown, you can see the German flag on the plane. But you also see an "N-xxxxx" registration which stands for USA. If it were a real German plane, it would have a "D-xxxx" registration.
The Equation - S1-E8
Revealing mistake: The shots of Peter and Walter entering St. Claire's are recycled shots from the pilot. You can tell because their clothes change.
Plot hole: When interrogating Boone, Olivia shows him a photo of a body with the spine torn out in the woods. This body hasn't been found yet, as a later scene has Olivia and Peter go to the same location and find the body. You can tell it's the same body because the shirt is the same in both the picture and the crime scene.
Inner Child - S1-E15
Continuity mistake: This is when Olivia goes to visit the boy at the hospital the second time. She and the guy from Department of Social Services go outside to talk. One of the shots is flipped. Note Olivia's notepad and the heart monitor by his bed. (00:19:40)
Continuity mistake: At the very end of the episode when Nina Sharp exits the elevator, she mostly steps out of the elevator completely before the masked men come and shoot her. However, when they review the security footage of the attack in the next episode, "There's More Than One of Everything," you see that she barely takes a step before being shot. (00:49:15)
Continuity mistake: This episode takes place over one day. It's Olivia's birthday at the start of the episode, and at night, she gets the card from her stepfather, implying that it's still the same day. Right before Broyles yells at Olivia in the Harvard halls, there is an establishing shot of Harvard where it is nighttime outside, but inside the building, there's sunlight coming through the windows.
Continuity mistake: On the park bench, while talking to Peter, Olivia's hair moves back and forth over her right shoulder. (00:32:40)
In Which We Meet Mr. Jones - S1-E7
Visible crew/equipment: During the shot where Walter is talking about Jimmy Hoffa while installing a light bulb, you can see the set and crew reflected on the metal part, including the audio coordinator with a boom mike.
Other mistake: When Peter is watching Richard Steig in the interrogation room and gets up to go in there himself, he tells Walter to stay there. On the live video screens behind him, you can see Peter enter the room before he actually does. (01:03:50)
Visible crew/equipment: When Olivia spots John taking off in his SUV, the reflection of a crew member's head is visible when she opens the driver's side door. (01:13:35)
In Which We Meet Mr. Jones - S1-E7
Deliberate mistake: Dunham comments that she works for Homeland Security. While the FBI is under the jurisdiction of the Justice Department, she works as part of a special Homeland Security task force; so saying she works for DHS is an accurate statement.
Continuity mistake: The front license plate on John's SUV falls off during the chase with Olivia. It reappears seconds later. (01:15:30)
Continuity mistake: When Olivia crosses over to the other side you see people wearing FBI jackets. But according to later episodes that had changed to Fringe Division on the other side years earlier.
Continuity mistake: At the club, when Peter and Olivia are looking for Valeria, Peter is wearing an ear piece in his left ear. When he talks to the woman, the piece is missing.
Power Hungry - S1-E5
Visible crew/equipment: When they're transporting Joseph Meegar out of the building and into the ambulance, you can see the reflection of the boom mike on the open car door, (the car Charlie is putting Mr. Fischer into). (00:41:00)
Factual error: When Walter removes the glass disk from Bowman's hand, he asks for a forcep, but is given a hemostat, which he uses to pull out the disk. My mother is a doctor, and when she saw this scene, she burst out laughing.
Inner Child - S1-E15
Continuity mistake: Olivia, Broyles, Peter and Walter are visiting the boy for the first time at the hospital. A small bit of Olivia's hair is in front of her left shoulder. Cut to the immediate next shot from the patient's room and all of her hair is behind of shoulder. (00:07:05)
Visible crew/equipment: During one of the shots when they corner Mitchell Loeb at the phone booth, you can briefly see a reflection of the entire crew on one of the cars.
Visible crew/equipment: At the very beginning of the episode when Susan Pratt is calling for a taxi, you can briefly see the reflection of a boom microphone on the passing taxi.