When Sister Benedict is starting to erase her drawn portrait on the blackboard when Father O'Malley walks in, she barely erases any of the portrait, but in the next scene most of the portrait is already erased.
Sister Mary Benedict (Ingrid Bergman) buys a training manual entitled "The Art of Boxing" by Gene Tunney. In reality, Tunney, a legendary prizefighter, never wrote such a book, although he did write two autobiographies.
Throughout the film, Sister Benedict is clearly wearing make-up and her eyebrows are drawn on, which is something that nuns would never do.
As the characters walk from the school building to the nearby church, they cast two shadows on the ground on both their right and left-hand sides, revealing that it is, in fact, a studio set illuminated by multiple overhead electric lights. In an actual exterior scene there would be only one light source overhead - the sun - which would cast shadows in one direction only depending on its position in the sky at the time of day depicted.
Fr. O'Malley tells two nuns he grew up and went to school in Missouri. In "Going My Way" it was established he went to high school in East St. Louis, IL.
In her initial encounter with Father O'Malley, Mary Gallagher tells him that her daughter's name in Patricia. Halfway through his initial encounter with Patricia, Father O'Malley starts calling her Patsy, not knowing that it is the informal name she indeed uses (as opposed to Pat, Tricia, etc.) as it had never been mentioned to him.