(at around 23 mins) This was the year of 1964 and Sister Aloysius Beauvier was going to decorate the classroom with a portrait of the Pope Pius XII. Then Sister James said that this Pope died "last year" (1963). Pius XII died in 1958. And the Pope that died in 1963 was Pope John XXIII.
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," which was introduced in 1970, not the Latin "Breviarium Romanum" that was in use in 1964.
When Sr Aloysius and Mrs Miller are walking through the local housing development, Parkchester, there are air conditioners in and awnings on the windows. In 1964, there were neither air conditioners or awnings on Parkchester windows. Air conditioners were not allowed in Parkchester windows until 1998.
(at around 24 mins) 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, which makes them contemporary shoes. In 1964, they would've had a black heel tag with white writing.
If this really took place in 1964, the text of Father Flynn's Breviary (his black prayer book also known as the "Liturgy of the Hours") would likely be in Latin and not in English as depicted in the movie.
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 trimmed short in that scene.
(at around 3 mins) The choir sings at the beginning of the movie. The film shows an outdoor scene, then (while choir is still singing the same song) an empty choir loft, then the boys singing in the choir loft.
(at around 39 mins) When Fr. Flynn and Sister James walk into Sister Aloysius's office, there is a lamp behind the principal's desk, turned on. Eight minutes later, the light is off even though no one touched the lamp.
(at around 1h 1 min) When Father Flynn sits down on the bench outside next to Sister James, he shows her flowers in his book. He moves the yellow ribbon book marker in between 2 pages, but in the next camera shot, the yellow book marker is crossing one of the pages.
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 14 mins) Sister Aloyisius confronts a student about earphones in class. Stereo earbuds didn't exist in 1964, but the monaural version she confiscates did (they were also used in hearing aids).
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 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.
(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.