Toward the end of the film, when Terry is on the sofa talking to Michel, she offers a cigarette to him. He picks it up, puts it in the mouth a little while and takes it out. Next shot, the cigarette is in his mouth.
When the sailor takes the photograph of Terry and Michel, their position on the railing changes depending whether the view is from in front (close together, nearly touching) or from behind (about two feet/ 60cm apart).
On the ship, at the railing, before and after disposing of the photographer's film of a compromising photo, shadows of Dunne are visible on the background screen of the ocean.; a shadow of her head before the photo incident, and her arm as she stretches, afterward.
When Michel is waiting for Terry on the Empire State Building into the evening, and a thunder storm breaks out, the same forked lightning strike is seen at least 4 times in the background, always in the identical location - which is highly unlikely.
When Irene Dunne and Charles Boyer are on the ship during the day and they are next to the railing talking and reading their letters, the ocean is moving behind them. Just after the close up on Dunne, she turns and faces the ocean, but you can see the shadow of her head on the screen that displays the ocean.