Continuity mistake: As Norton kills himself, the glass that blows out behind him is quite large and about half of the frame gets blown out. There are a few specs of blood on the window as well. A couple of shots later as the camera pulls back and shows Norton you can see the hole in the window is quite small and shaped differently and a smear of blood above the hole now.
Continuity mistake: Andy is discovered missing. While the warden is walking to Andy's cell, he tells the guard to question "him", meaning the Morgan Freeman character. He points to Freeman with his left hand but when the scene changes its point of view he is pointing with his right hand.
Continuity mistake: When Norton prepares for his suicide you see on the desk a photograph and, a bit away, a lamp. After he has shot himself the photograph is leaning against the lamp (it was too far away in the previous shot to have fallen against it). There's some sort of card open on his desk on some paperwork which suddenly moves onto bare desk as well.
Visible crew/equipment: When Sgt. Hadley nearly throws Andy off the plate factory roof, the reflection of a boom mike is briefly visible in the shiny brim of Hadley's hat. Andy dropped the tar mop before approaching Hadley, and there are no other objects in this shot that could have caused the reflection.
Continuity mistake: When one of the "sisters" gets beaten up by Capt. Hadley he tries to escape from his cell and holds on to the bars on the other side of the passage. From the side his right hand grabs a diagonal bar, but seen through the bars he is holding on to a vertical bar.
Continuity mistake: When the warden is given a bribe by his contractor/buddy, it comes in a box containing a pie and the pie is in a disposable corrugated aluminium pie tin. The warden gives the pie to Andy in a later scene and when Andy and Red eat it, it is in a metal pie pan... the non-disposable kind.