Early in the film, a co-worker attempts to discuss the film The Quiet Man. This scene in Brooklyn, NY, takes place in 1951; also in a key scene that takes place much later, a new tombstone on a grave is dated 1st July 1952. "The Quiet Man" was not on general release in USA cinemas until 14 September 1952, with 21 August 1952 American premiere in New York City, New York.
When Eilis first goes through US Customs you see a far shot of her lifting her suitcase onto the counter for inspection. Then in a subsequent shot you see Eilis standing second in line (one person is in front of her), and then it cuts to her standing in front of the Customs agent and getting her passport stamped.
After the scene at Coney Island, Eilis does a voice over of a letter to her mother. During that voice over, in a pan shot inside the department store where she works there is a mirror in the background where you can at one point clearly see the camera that is shooting the scene.
When Eilis is on the deck en route to New York she is facing the sun, which means she is facing south. Yet the ship appears to be moving from right to left in the water. That means it is moving in an easterly direction, in other words, away from the USA.
When you first see Eilis at the stoplight intersection, going to work she is in a crowd of people on the street corner. In a much later, brief shot of her walking to work again you see the very same people in the very same places at that stoplight intersection. They no doubt used the same footage for both scenes.
The goof items below may give away important plot points.
As they leave Nancy's wedding, the incidental music coming from the church is Pachelbel's Canon in D. While it was composed in the late 17th century and is frequently played at weddings today, it was an obscure and almost forgotten piece in 1952. It only became famous when a new arrangement, tempo and some additional parts were added to it for a 1968 recording. Even in the unlikely case of someone playing it at a wedding in a small Irish town in 1952, it would have sounded nothing like what we hear in the film which is the modern version we play today.
On her second trip to the U.S., Eilis stands in the returning citizens line, although she could not possibly have been a U.S. citizen at the time. Naturalization requires a minimum of five years, and, since 1922, marrying a U.S. citizen does not automatically confer citizenship on the wife.