Factual error: When Red is sitting under the oak tree in 1967 and pulls the money out of the envelope, the top bill is signed by Nicholas F. Brady, who was Secretary of the Treasury September 1988 - January 1993. It's less about reading the handwriting as noting the different appearance of different signatures from different eras.
Factual error: Towards the end of the movie when Andy is escaping, he breaks open a sewage pipe and is covered with its contents. It is physically impossible for the sewage to shoot out of the break like it did. In order for that to happen there has to be pressure. But there is none there, because the end he crawls through is empty and the end is in the open air.
Plot hole: When the warden comes into the cell the morning after the escape, the poster covering the hole is fastened down on all four corners - impossible to do after squeezing into that small hole. [On the DVD commentary, the director confirms that this was a movie "cheat".]
Continuity mistake: When Andy is playing the opera music, the whole prison stops what they're doing and stares at the speakers. When they show the infirmary, an African American man gets out of bed and looks out of the window. When the camera goes to the prison yard it pans around and goes right across the same African American man, who is now standing in the yard.
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.
Factual error: In the library, Heywood is listening to Hank Williams' "Lovesick Blues." He is playing the double LP "Hank Williams 24 Greatest Hits," which was originally released by MGM in 1970. "Lovesick Blues" was recorded long before this, but it is not possible for him to have acquired that particular 1970 double LP.
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: In the scene where Brooks Hatlen is committing suicide you see him engraving "Brooks was here" into the wood panel near the ceiling. The table he is standing on puts his face opposite the vertical wooden bars underneath the engraving (we see this in a shot looking up at him through these bars). Then when he rocks to tip the table over to hang himself he doesn't seem to fall at all, maybe a few inches if anything, yet when we see the pan out he has fallen at least 2 feet.