When Father O'Malley is talking to Jenny while she is getting ready to perform, the conductor can be seen taking out his cigarette case and opening it twice, in two adjacent shots.
After seeing the bishop, Father Fitzgibbon calls Father O'Malley into his office, where they have a long talk. There's a clock in the background, often seen over O'Malley's shoulder. From the first and last time the clock is seen, about three minutes go by, but the clock always says 5:35.
When Fr. Chuck says to Fr. Fitzgibbon, "She's a singer", Fr. Chuck's hand change positions.
It was common custom at the time for opera singers to be classified as either soprano or contralto (alto). The term mezzo-soprano only gradually became more widespread from the 1950s onward. Now, of course, she is more correctly known as a mezzo-soprano, but in the 1940s, she would have been referred to as a contralto.
Father O'Malley apologizes for the window broken by a baseball and offers to pay for it, but he wasn't the one who broke it.
However, as a priest, he's simply practicing the trait of charity. Being new in town, he likely would want to make a good impression, especially on an atheist.
When Father Fitzgibbon takes Father O'Mally on a tour of the Church, there wouldn't be an organist on duty unless there was a Mass in progress.
The organist wasn't "on duty" but probably was practicing or simply spending time in "God's house".
Ted Haines Jr. tells Father O'Malley that "with you it's different. You don't have anything and you don't want anything." In reality, the Roman Catholic Church is the wealthiest religious organization on Earth.
While the church is very wealthy, that doesn't translate to individual priests being wealthy. Priests in some orders even take a vow of poverty.
It is unlikely that Father O'Malley would be transferred to a new Parish without being told where it is located.
Actually, in that era, the priest likely was told only the parish's name and the town or city.
Barry Fitzgerald's golf grip is wrong. His left hand should be above his right.
Obvious lip syncing during "Going My Way".
In Carol's apartment, when father O'Malley enters, a crew member's hand can be seen closing the door behind him.
Directly after the first rendition of "Going My Way", the shadow of the boom mic can be seen moving on the church wall behind father Fitzgibbon and Miss Linden.
Shadow of mic visible behind Ted when Father O'Malley goes to Carol's apartment.