While the mother is drinking from a bottle of vodka, the level in the bottle goes from 25% full to nearly full and the shape of the bottle changes.
When Chris Pines' Character is in the phone booth speaking with his editor, his camera is shown on the shelf in the phone booth. After a few cuts, he leaves no camera on the shelf and he does not take it with him.
About 30 minutes in when Fauna is learning about her birth certificate the tea cup her mother puts on the shelf disappears then reappears in the next shot.
Pat/Fauna calls Los Angeles and speaks with the grandfather, George Hodel, in a scene set between 1960 and 1970, when Ford Falcons were in production, a car Jay Singletary is driving in the episode. LA Police records have George Hodel moving to China in 1950, to avoid another murder investigation, not returning until 1990.
At 36:05 Jay (Chris Pine) is pulled off the hood of the unmarked police car. As his back is exposed, the handcuffs that supposedly are restraining are not latched. The single strand (the part that swings around and locks) is hanging free on the left hand. The right cuff appears loose or not closed as well.
The Greyhound bus Fauna takes from Sparks, NV to Los Angeles, CA is a transit bus, with two sets of doors, designed for city use, not intercity routes.
Dr. Hodel's trial took place in 1949. There is an exterior shot of the Parker Center which did not become the headquarters for the LAPD until 1952.
Two of the three ambulances at the hospital Jay arrives at are from 1969 and 1967. The mini-series takes place in 1965.