Grass on the Streets and Weeds on the Rooftops - S4-E10
Plot hole: Beth surprises Jamie in the act of disposing of his father's body at the train station. But Beth doesn't learn about the train station until season 5, episode 8. So, how did she know where to catch Jamie in the act?
One Hundred Years Is Nothing - S5-E1
Continuity mistake: How is it that Carter has grown so much in 9 months? He looks 2 years older.
Continuity mistake: Beth offers Rob a cigarette and places her cigarette case on the table, setting her gold lighter upright next to it. Although Rob doesn't immediately accept her offer, the lighter's position changes in subsequent shots—first standing upright, then lying flat on the table, and finally back to being upright again. (00:49:40)

Cigarettes, Whiskey, a Meadow and You - S5-E6
Continuity mistake: When Mo arrives at the rally, he shows his badge to a soldier. He either holds it from the upper or the bottom part, depending on the shot.
The Apocalypse of Change - S5-E10
Other mistake: S5 E/0: The Apocalypse of Change - Tate is wearing a ball cap with a buffalo on the front, which would be fitting since his family's business is cattle ranching. Since the Dutton family are proud residents of the State of Montana, it's strange that Tate would be wearing a ball cap with the State flag of Wyoming. (00:15:08)
The Apocalypse of Change - S5-E10
Other mistake: While in Amarillo, Texas, Rip tells Beth that it's the first time he has ever been out of Montana. However, since the "Train Station" is located in a rural county in Wyoming and is a secret place where bodies are disposed of, Rip cannot admit that he has been to Wyoming. (00:21:54)
Continuity mistake: Sarah goes to Jamie's home after a difficult day, and Jamie comes in shortly after he has been "rescued" by the new governor. After a terrible fight, Sarah goes over and grabs an empty glass from the top of the bar. The glass is suddenly filled with ice when she drinks the vodka.
Continuity mistake: At the end of the episode, when he does the carjacking, the car pulls away, and the car in front has the same licence plate as the car they are in.