When Detective Loki is speeding to the hospital at the end, he passes the same restaurant (with a red awning) three times: once as he pulls off the smaller road onto the major road, and twice after that.
When Detective Loki goes to question Bob Taylor at his home (the house with the snakes), the mailbox pole by the driveway shows the address number as "4261" but when Loki calls for backup he gives the address as "427 Carol Street".
When Loki leaves Keller's house and is sitting in his car about to leave, he has a toothpick in his mouth. Keller runs out of the house to speak to him. We see Loki from inside the car as he opens the car window and talks to Keller while still chewing the toothpick. In the following shot from outside the car the toothpick has disappeared.
At around 4 minutes in the movie, after the grace says "wait for your mother" she walks down the steps towards the car, but in next scene she's shown walking twice down the same path just past the steps towards car.
The area code on Dover's truck is 570, which is the Scranton/Wilkes Barre area of Pennsylvania, but the license plate on Loki's car indicate Montgomery County, which is a couple of hours south and does not use the 570 area code.
Detective Loki works for a municipal police department in Pennsylvania yet all of the police cars with lights activated have only blue lights. Blue lights in Pennsylvania are "courtesy lights" for volunteer firefighters/EMS personnel private vehicles. They are not emergency lights with any traffic control powers. Police use either a combination of red and blue or, if using a single light, red only.
When Keller is deer hunting with his son, the two families are celebrating Thanksgiving. The film takes place in Pennsylvania, where deer hunting season does not begin until the first Monday after Thanksgiving.
When Keller Dover (Jackman) is smashing the bathroom sink, you can see that he's using a prop hammer. When he holds his arm up, the handle near the "cheek" of it bends. It then straightens itself out after he puts it into the wall next to Alex Jones' (Dano's) head.
When Detective Loki is on the computer looking at the Obituaries a little over halfway through the movie, the taskbar of the computer can be seen, and it shows the symbol for no network connection (computer with red 'X') even though he is obviously supposed to be online.
The goof items below may give away important plot points.
When the suspect spent hours drawing a maze, Loki who is watching thru the glass asks another officer for a key card to go back into the room. As he is beating the suspect, the other officers run into the room and the outside of the door only has a doorknob and no lock.
Possible plot hole: how did Keller Dover find his daughter's missing red whistle in the pit beneath the Trans Am? if she had only just lost it before the Thanksgiving party which was before she was even kidnapped? It would have gotten there before she even reports she lost it?Chronologically impossible. It is explained in the movie, that both girls found it before the kidnapping. The kidnapper hide the girls down there when the police showed up and the girl must have lost it then.