When searching Rudy's apartment, Dexter goes over to look at the Barbie record player. The African-American police employee who is cataloging the items is clearly wearing a laminate with a photo of a bearded, white, middle-aged man on it.
When Dexter and Doakes have their confrontation in the shipyard, Dexter is seen placing his tongue between his upper and lower incisors. No one trained in any sort of fighting technique would do this as it incurs the risk of a blow to the chin severing the tip of the tongue.
Dexter finds the story of Laura's murder in the October 3rd edition of the newspaper. The October 3rd paper would report on what happened October 2nd. Brian would have known that they were actually "born free" on 10/2, not 10/3 (leaving aside the fact that if they were found on 10/2, then the murders actually occurred on 9/30).
The goof item below may give away important plot points.
Crew or equipment visible
At the end of the episode, when Dexter flips the switch sending Rudy Cooper/Brian Moser upside down, the tube pumping blood out of his neck is clearly visible to the left running up the metal frame and out the back right side. Even a piece of black gaffer tape is seen in the second/final shot stuck to Rudy Cooper/Brian Moser's chin.