Episode #1.6 - S1-E6
Continuity mistake: Sami takes Maya directly from the location where Tommy Dark's body is found to his home, but Maya's outfit is completely different when they arrive.
Continuity mistake: This show has been widely declared as canonical with the games, by the makers of both this show and the games. The game Fallout 4 establishes the date and time that the bombs drop: Saturday, October 23, 2077, 9:47 AM Eastern Time. Which means it was 6:47 AM West Coast Time. That's a pretty weird time of day to schedule a kids' party with lots of guests and a hired celebrity cowboy.
Suggested correction: They changed a lot of subtle things from canon for the show. Also, all games have got some inconsistencies between them as well. In the show, on the TV, there is a clock that says it's around 9:30 AM the first time we see it and 10:45 when they cut the cake, so it's close to the afternoon and not very early in the morning. The position of the sun confirms this.
Second Date - S1-E2
Continuity mistake: At the deli, Jane orders plain bagels with cream cheese. When she gets home, she has everything bagels with lox.
Episode #1.1 - S1-E1
Continuity mistake: When Emma gets a pen out of her bag so that Dexter can write his number, the pen is upright. When the shot cuts, it is upside down, then it is upright again in the shot after that. (00:35:04)
Paul Is Dead - S1-E6
Factual error: The female British astronaut is in her space suit. There is a supposedly British flag on her left sleeve. But the flag is wrong. The red diagonal stripes are centered on the white field. They should be off-centered by 3/5 of the white space. Not centered. This is a common mistake you find in British flags bought outside the UK.