Character mistake: The team identifies a blue Lexus fleeing the scene of the murder, but later in the show OA says to Maggie that they've located the black Lexus.
Plot hole: The FBI is looking for a "very tall" guy who has been identified as such by various witnesses and by their own expert due to the entry angle of a fatal bullet wound - 6'5" at least. They send Elena into a bar near the end of the show to meet the Albanian perp who when he arrives is barely average height.
Undisclosed - S2-E7
Continuity mistake: Agents follow the suspect after the shooting in an apartment. They chase him upstairs and out of the door but are still at street level.
Continuity mistake: The chase scene toward the end of the episode is a mess. Toward the end of the episode, Scola runs through a door twice while chasing the suspect and, just a few scenes later, Tiff, while chasing another suspect approaches another door twice before she enters.
Continuity mistake: Toward the end of the show Scola and the brother of the principle terrorist emerge from the building containing the now disabled bomb. The brother and Scola exchange a few words and the brother walks away. After a very quick camera change the next shot shows the brother far away from Scola, much too far for him to have walked given the shortcut away.
Continuity mistake: Nick drags Nathan out from the silo where they are hiding twice.
Continuity mistake: The scenes of OA and Maggie chasing McGowan are a mess. The distances between the agents and the perp vary greatly from moment to moment and OA is going over the same fence twice.
Continuity mistake: The two FBI agents are searching for the wounded perp and track him to a veterinarian's-the woman working in the office is ordered to the ground and lays down in front of office chairs very close to large animal cages by her head. In the next scene when she is ordered to her feet, the cages are nowhere near her.
Liar's Poker - S3-E3
Other mistake: Her name in the show is Elise Taylor, but on a badge in her car it's Elise Smith.
Factual error: The digital time zone clock on the wall in the FBI operation center is wrong. The show is set in NYC, which is Eastern Time. The clock shows EST at 16:44 and ZULU time (UTC / GMT) at 20:44. The difference between EST and ZULU time is 5 hours, so the clock should show ZULU time as 21:44. Plus, the times for the other time zones on the clock are wrong by an hour. (00:18:29)
Undisclosed - S2-E7
Other mistake: Toward the end of the episode the perpetrator is corned by the FBI. He steps backwards from the curb and is killed instantly by a garage truck moving at a high rate of speed. In the following scene the garbage truck is shown just a few steps beyond the scene just behind a parked car. The truck was going very fast when it killed the perp and could not have stopped so quickly. Further it would not have been going that fast in a curb lane with a parked vehicle right ahead of it.
Compromised - S1-E9
Character mistake: When she and OA meet Marshal Ackerman, Maggie is somewhat surprised to learn Damon Cruz was a federal witness. As a federal agent herself, Maggie surely would know that the Marshals Service's mandate includes the witness protection program.
Continuity mistake: Halley has several abrasions and cuts on her face and lip. The next time she is shown her lip is healed perfectly but all of the other abrasions and cuts are still visible. (00:10:00 - 00:23:00)

Revealing mistake: The FBI woman picks up the phone to speak to her boss but the phone is off. (00:22:38)