Revealing mistake: Episode 15, "Faceoff": When the agent fires the hook to get to the roof of the church, it does not reach the balcony. However in a following shot you can see the agent climbing there. (00:10:05)
Continuity mistake: Episode 15, "FaceOff": Sophia gives Elias one hour to get the buses or else she will destroy Washington. When this happens, the sun is high, and by the time the buses arrive it is already very dark outside. (00:03:05 - 00:34:15)
I Haven't Told You Everything - S1-E1
Continuity mistake: In the first episode, the pilot of the plane's shoulder rank changes from stars to stripes, back and forth, repeatedly from scene to scene.
Other mistake: In episode 5, while Sean is talking to Leila on his iPhone, it is obvious that there is no connection, because Home screen of iPhone is displaying on device. (00:14:20)
Protect Them From the Truth - S1-E3
Visible crew/equipment: When the RV hits the police vehicle and sends it rolling down the road, you can see the reflection of the crew on the back bumper of the car as it rolls to a stop.
Plot hole: Episode 1: All passengers on cruise ships have a passenger card with their name and cabin number imprinted. They are not just identical key cards like in hotels.
Protect Them From the Truth - S1-E3
Visible crew/equipment: During a shot near the crashed plane, the camera and the crew can be seen in the reflection of the president's glasses. (00:10:55)
Visible crew/equipment: As Thomas watches the plane take off into the air, a car leaves, driving by him. As it does, you can see the cameraman reflected on the car.