As the ocean liner carrying the cast passes the Statue of Liberty, the name on the bow is "Rotterdam". Later, it's referred to as "the Northampton", and even later, as the "S.S. Britain".
Near the end, Charles, Lord Dilling, and Fay are sitting in her room after receiving the call from Scotland Yard. Montgomery offers cigarettes to the other two. In the further shot, Montgomery is several feet from the other two, but in the closer view when he lights the cigarettes, he is very close to them.
After Fay Cheyney has a small visit from Joan and goes to relock the door, the scene changes slightly to another take from that moment forwards. As she goes toward the door, there is a cigarette in her left hand. After closing the door, the cigarette has disappeared.
Standing on deck, Fay (Joan Crawford) is trying, unsuccessfully, to light the cigarette in her holder with a lighter. She is seen by Willie (Nigel Bruce), who lights it for her. Later, when offered a cigarette by Lord Dilling (Robert Montgomery), she states, "....you know I don't smoke." Since the characters have speculated about her, Lord Dilling likely would have known that his friend lit her cigarette aboard ship.