When Fr. Flynn and Sister James are on the bench outside the school, Fr. Flynn opens his breviary. It's an English "Liturgy of the Hours," introduced in 1970, not the Latin "Breviarium Romanum" used in 1964.
When Sr. Aloysius and Mrs. Miller are walking through Parkchester, the apartment windows have air conditioners and awnings. In 1964, no Parkchester windows had air conditioners or awnings. Air conditioners weren't allowed in Parkchester windows until 1998.
Father Flynn turns during the basketball scene in the gym, showing the backs of his Converse Chuck Taylor All-Stars. They have a white heel tag with black writing. In 1964, they would've had a black heel tag with white writing.
When Viola Davis walks away from Meryl Streep, cable TV amplifiers are clearly mounted on the outside building wall. In 1964, cable television service was only available in Manhattan. New York's outer boroughs got cable TV in 1986.
When Father Flynn sits down on the bench outside next to Sister James, he shows her flowers in his book. He puts the yellow ribbon bookmark in between 2 pages. In the next camera shot, the bookmark is crossing one of the pages.
As Father Flynn sits alone in Sister Aloysius' office after their final confrontation, he handles his breviary and arranges two pressed flowers on its cover. His notoriously long fingernails are suddenly short.
Sister Aloysius picks up a ballpoint pen. Sister James says she doesn't let the children use them. Sister Aloyisus says "And yet, here one is." In the next shot, she walks and speaks a few feet away from where she was already standing, with her line of dialogue starting in the previous take, but her lips aren't moving.
During Mass, a song is sung before and after the Homily (sermon). In a Catholic Mass, the Gospel reading is right before the Homily, and the Profession of Faith (Nicene Creed) is immediately afterward. The bit at the end of Gospel and right before the Creed may be sung or chanted, but they would not sing a hymn.
During the sermon, the burse (the rectangular object on top of the chalice veil on the altar) is incorrectly placed. At a Low Mass in 1964, the burse would have been removed from the chalice at the beginning, and placed to the left of the chalice, leaning upright.
(at around 1h 35 mins) In the final scene, Sister James walks up to the seated Sister Aloysius. When the camera zooms out at the very end, there are no footprints in the snow on the sidewalk. The snow was added in post-production.
In the movie, the street signs have black letters on a white background, without cross-streets. In 1964, NYC street signs had white letters on blue enamel backgrounds, with the cross-streets printed in smaller letters above.
The classroom door opens inward. Ever since the 1908 Collinwood school fire in Cleveland, laws have required classroom doors to open outward. It was initially believed that the panicking children pushed against the doors, which prevented escape from the fire and resulted in 172 deaths. However, later investigations proved that the doors did in fact open outward. The school was rebuilt and renamed Memorial school. It's surrounded by gardens in memory of the children who died there.
(at around 18 mins) William is sent home for having a bloody nose. He walks through an intersection with crosswalk indicators. The traffic signal changes from red to green, but the crosswalk indicators don't change.
When Sister James talks to Donald after he visits the rectory, the young actor is not Joseph Foster. Neither actor's face is visible, but the student's hand is white. Donald is the schools' only black student.