In the establishing shot in the nightclub, where Grace tells Gus she's pregnant, she's seen snuffing his cigar out in ashtray. In next shot, he's smoking it, then she yanks it out of his mouth and it extinguishes it a second time.
When Grace tells Gus she is expecting their second child at the end of World War in 1918, this could not happen because their second child was not born until 1922.
There is a noticeable tan line on Danny Thomas' ring finger on his left hand when he removed his actual wedding band while appearing in all his scenes before he is married to Doris Day's character.
There is a shot of the Empire State Building prior to the stock market crash of 1929. Construction on the Empire State Building started in 1930.
Shortly after struggling Gus Kahn moves to Hollywood following the 1929 stock market crash, someone makes a remark that Kahn will know he's made if he can afford a table at Romanoff's restaurant. In reality, Romanoff's didn't open until 1939, just two years before then well-established Kahn died.
When Gus first visits New York in the mid-1920s, the establishing shot of Times Square includes many early '30s car models as well as a theater marquee listing a film first released in 1933.