The position of the spoon in Henri's hand changes while he's at the table just before murdering Rufus.
During the first meeting between Young and Stamphill in the open holding cell, the guard in the background disappears and reappears between shots.
Henry Young didn't go to Alcatraz for "stealing 5 dollars." In real life, he was both a bank robber and a murderer, and had already been in at least two state prisons before landing at Alcatraz.
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The judge schedules Henri's trial for "June 22, a week from Thursday," in 1941. June 22 was on a Sunday in 1941.
Although the movie took place in the late 1930s, all exterior shots of Alcatraz show the burned out shell of the warden's house. The warden's house on Alcatraz didn't burn down until 1970, seven years after it was closed as a prison, during the Indian Occupation of the island.
Along with the warden's barracks, exterior shots of the island also show the rubble on the south end of the island when the apartments where many of guards' families should be seen, given the time period.
The Bradbury Building is in Los Angeles not San Francisco so it's more than a cable car ride to it.
All of the street cars aren't running on tracks as they are in fact trucks as the tires can easily be seen from several different angles.
In the scene when Christian Slater's character is running down the hill after the Trolley in San Fransisco, you see many cars that are appropriate for the period lining the streets. If you look closely, however, you'll see Red-Zones. There weren't Red Zones in San Fransisco during this time.