Factual error: When the group are heading to California after leaving the Missouri home of Lt. Warren they somehow end up in South Dakota. And go within 251 miles of Salt Lake City and then back track over the Rocky mountains to the facility in Fort Collins Colorado.
Revealing mistake: At the end of the episode when they show Citizen Z, the computer on his left has data or some kind of code running down the screen. Right after he puts his glasses on, the pause button appears mistakenly on the screen, revealing that it's not legitimately a program running, but just a video the effects team setup.
Continuity mistake: Right at the end, after Cassandra throws away the finger, there is the scene with her "family" finding the music box. Cutting back to her; she again has the finger resting on her shoulder/hair.
Philly Feast - S1-E3
Continuity mistake: Cooking steaks on the barbecue, one for him and one for the dog. On the barbecue, they are T-bones, but when he passes one of the steaks to the dog, it is a boneless cut. (00:37:00)
Other mistake: They have a great electric fence, but the power goes out because of the storm. They all panic and flee the house, but the lights are still on in the house.
Revealing mistake: As the team is headed to Warren's hometown the show shows a clip of zombies in the field but it's the same 12 people (exact same outfits too) used in 11 different places in the exact same shot.
Audio problem: 10k pulls a knife to stab a zombie through the fence at the nuclear plant but they dub in the sound effect for a gun when he stabs the zombie.
Die Zombie Die... Again - S1-E10
Revealing mistake: When Murphy gets back to the group after having walked about 10 miles with Amelia following who is now a walking Z, why doesn't she run past Murphy to the others?
Character mistake: Doc holds the binoculars the wrong way round. (00:30:20)
Revealing mistake: The lady that is with the other two guys that kidnapped Murphy decides to remove her magazine from her pistol. It's an airsoft pistol. The magazine has a front feeding area for 6mm BBs, and looks weird in general. She also has no ammo, as you can see the entirety of the internal spring - not bullets. And the magazine is weird too.
Other mistake: About 10 minutes in, Doc cocks his shotgun. Then he does it again, and no shellcase is ejected.
Answer: When the plant zombies would attack they would wrap people with the vines and hold them, so they couldn't escape.