On the boat ride over to the prison from San Francisco a radar unit can be seen spinning on top of the boat. That type of radar transponder wasn't developed until the mid 1970s. According to the information on screen, this took place in 1960.
In his opening monologue, Warden Madigan mentions his Predecessor's Johnston and Blackwell, when in fact he should have said, "Johnston and Swope". This film was set in 1960 and Olin G. Blackwell did not take over as warden until 1961.
When Morris meets the warden on his arrival, the warden states that no newspapers, or magazines carrying news, are issued to the prisoners. However, Morris' library duties later include delivering issues of The Saturday Evening Post and Ebony, publications containing news items in each issue.
In his opening monologue, the warden mentions his predecessors, including a previous warden named Johnson. He's presumably referring to James A. Johnston (note the T in his last name, which the warden does not pronounce), who was the first warden of Alcatraz, presiding over its opening in 1934 and retiring in 1948.
When at the warden's briefing with Morris, the warden drops his nail clipper into the dish with another nail clipper present. At this point they both are closed. At the next shot of the dish after Morris removed one nail clipper, the remaining clipper is now all of a sudden open.
In the beginning, where Frank Morris is being escorted through Alcatraz, we see him from behind, naked. We also see him being escorted by the guards from within a cell, and very brief, you can see that he is wearing a white towel around his hips. In the next shot, he is seen naked from behind again. Finally when being put in his cell, he is once again wearing a towel.
When Doc is painting in the prison yard, the spots of wet paint on the left-hand side of the canvas change. Doc and Frank are having a conversation - the shots showing Frank speaking do not show the canvas for the most part, while the ones that show Doc speaking do. The wet spots increase and decrease in size and number every time the canvas is shown.
The goof items below may give away important plot points.
In the final scene when the warden is looking around Angel Island (after the escape) on a shot of San Francisco from Angel Island, you can see the Pyramid Building or Trans-America Pyramid, not built until the 1970s.