When we first see Abby's revolver in her purse, it is a detective-style gun with an enclosed hammer and no cocking lever. For the rest of the movie, it is a standard revolver with a cocking lever - a completely different gun.
When Ray and Marty are on the dark road, Ray is driving a 1966 Buick Skylark, but when he shifts gears we see a shot of a dashboard with a round speedometer marked in both MPH and KM/H. '66 Skylarks have a horizontal speedometer and would either be marked in MPH (USA) or KM/H (Canada), but not both.
When Detective Visser wants to remove the knife from his right hand, he tries to do it with his left hand "reversed", but in the next shot he removes the knife with his hand in a more natural position.
Loren discovers his lighter is missing when he goes to light a cigarette after burning most of the evidence (the doctored pictures). Logically, he would've used his lighter to light the pictures in the first place so he would've discovered his lighter was missing when he tried to burn them, not light a cigarette afterwards.
A radio station's call letters begin with W. The boundary in the US between radio/TV call letters beginning with W and those beginning with K was not always the Mississippi River. In the 1980s, Texas still had more than one "W" station.
When Visser (the detective) is sitting at the table in the bar across from Marty (the first time) he lights his cigarette with his lighter and a shot shows his hand putting the lighter on the table. The hand looks like that of a younger man, and the huge ring on Visser's right hand is missing in that shot.
The goof items below may give away important plot points.
When Abby tells Meurice she thinks Ray killed Marty, Meurice turns back into his apartment, saying, "Well, that's bullshit." His lip movements are clearly mismatched. (This is in the Director's Cut; the original version had Meurice saying, "All right, but I'll tell you right now I'm retiring.")
After Visser shoots Marty, he drops the revolver without cocking it, so the shell under the hammer is an empty one. He kicks the gun across the floor, still without cocking it. When Ray visits the bar later on, the gun somehow manages to cock and fire itself just from the pressure of Ray's feet on the floorboards.
When Detective Visser removes the knife from his hand, it is clearly visible that more than half level of the knife is covered with blood, but in the next shot where he picks up his hat, you can see that only the knife tip is covered with blood.
When Ray is beginning to bury Marty, as the camera tracks past the car (whose headlights are supposed to be the only source of light), a stage light is briefly visible on the other side of the car. It's cropped out of some video releases.