Marathon-Thon-Thon-Thon-Thon - S1-E10
Continuity mistake: In the opening scene Dan has a newspaper in his left hand. When he walks away it has moved to his right hand.
Two by Two - S1-E6
Continuity mistake: When Alicia presents the slingshot trajectory, Eva says the solar sails will provide "more than enough" power to reach the planet with the U238. Shortly after the sails are deployed, the edge of one of them is damaged, causing the ship to lose enough power to start dropping back to the star. But you can see that about 1/10 of one of the six sails is damaged, or 1.7% of the entire sail array. How is "more than enough" reduced to "not enough" when they lose such a small part of the sails? (00:19:57 - 00:23:33)
Continuity mistake: When talking on the roof at the start, the guy in green has his hands in his apron pockets when being told the security cameras are off. Cut to a wider shot and his arms are folded instead.
Continuity mistake: Hannah enters the bar wearing a University of Texas hooded sweatshirt with its unmistakable school color when she asks Charley to take her to see her father. When they arrive at her father's house, she is wearing a plain dark sweatshirt.
Deux Amours - S1-E5
Continuity mistake: Daryl fights the walker on the ship after it kills the other prisoner. He gets backed into a corner and jumps off the boat just as the explosions are happening. The walker he was fighting disappears.
Continuity mistake: As Jizzlord says, "so really, this is a good thing", he's holding the posters upright, facing Jen. Cut to a shot from the side and he's holding them horizontally, facing upwards.
Please Hold to My Hand - S1-E4
Continuity mistake: When Joel tries to drive through the rebel trap, the bad guy throws a big rock/brick on the pickup that creates a reasonably sized crack on its windshield - as we can see it from inside the car. However, during all the next outside shots, the windshield is intact - actually, it looks brand-new.