Episode #1.4 - S1-E4
Visible crew/equipment: When Lee is being interrogated, at one point (after the detective says "Are you crazy?") the detective walks towards the large two-way mirror and you can see a clear reflection of a cameraman in the lower part of the mirror. (00:27:31)
Character mistake: The Enterprise is targeted by projectiles from the planet. Spock identifies them as "plasma torpedoes", and Captain Pike reacts with surprise; "Plasma torpedoes? That's 21st century tech" and from that they all establish that the civilization they contacted is not yet capable of warp drive. Plasma torpedoes are the main weaponry of the Romulans centuries later, and Earth itself did acquire warp drive in the 21st century, so this claim and deduction don't seem to add up. (00:22:15)
Revealing mistake: When Jen is at the cash bar and keeps putting money down, it's the same shot used multiple times since it's the same serial number on the $5. And in one shot they flipped the image to make it look different (you can see the "5" is a reverse image).
Other mistake: The King's ship has no steering devices, steering wheel/pin, no steerman, and no crew. (00:26:00)
Continuity mistake: When Reacher meets Neagley in the New York diner, he takes two first bites out of his sandwich.
Factual error: Detective Superintendent Hamilton has red fabric within the crowns on her rank insignia. This would be correct for most British police forces, but the Metropolitan Police uses blue fabric instead.
Continuity mistake: Obi-Wan Kenobi exposes Haja Estree. He asks for his help. When he hastily says "Yeah I'll do it!" Kumail Nanjiani's right hand is curled in a loose fist or fully opened, back and forth as camera angles switch. (00:08:25)
Continuity mistake: Toward the end of the episode, our intrepid heroes stash their large black bags as they run through the market. They run through the market for the next several minutes with only their machine guns. They later board the waiting speedboat but when they disembark, they have their bags.
Factual error: The German series is set in 1987 in Berlin before the wall came down. It shows a suitcase going through airport security that wasn't set up until after 2001.