When Joe enters a ballroom searching for Emma, the press card in his hat is vertical. In the next shot, this card is leaning to the back of the hat. In the following shot, it's vertical again.
At the movies with his son, Joe muses about the rise of Hitler and suggests that it is unlikely there will be another war. As the credits roll, the production year is MCMXLI (1941).
When the bank is robbed, the cop who comes out fires 14 shots from a 1920s police revolver without reloading. AT the time, typical revolvers held only 6 shots.
In an early scene, someone sings "sugar in the morning." The song "Sugartime" was published in 1958.
Loretta is speaking at the church 1:20.19.. She has acrylic tips, not sure they had those back during the early 1900s.
In 1927, when Joe and Emma are in bed together talking, "Moonglow" is playing in the background. That song was written in 1933, and the music playing is a 1941 Artie Shaw arrangement.
When Chris Cooper's character is sitting in his chair repeating the word "Repent", the camera pans past a picture of Jesus. That picture, called "The Head of Christ", or The Sallman Head", was painted by Warner Sallman in 1940.
The opening sequence includes a photo of a dead Civil War soldier with a bloody face. Ben Affleck's character fought in World War 1.
Joe's father says that one of the policemen who fell in the lake died "of hypothermia". That's a technical term not used outside of the medical community until about 1975. In 1920, he might have said the officer died "of exposure" or even "of the cold."