Factual error: In the beginning of the film, when Nick flashes back to 1922, he mentions that the buildings were taller, and a shot of the Empire State Building is shown. But construction on that didn't even start until 1931. Even the Chrysler Building didn't start construction until 1929.
Factual error: The movie takes place in the summer of 1922. They played Rhapsody in Blue, which wasn't written till 1924. The bandleader is obviously Cab Calloway. Cab Calloway's first recordings with the Missourians weren't until 1928. "Oh! You Have No Idea" also wasn't recorded until 1928, and it was sung by Sophie Tucker, not Josephine Baker (although they do credit Ms. Tucker in the final credits).
Factual error: Trains of various types appear in several shots but the trains shown - likely all CGI - are completely unlike any trains that were in use in New York in the 1920s or in any other era. They appear to be based on Australian prototypes and their design differs significantly from American trains.
Factual error: The "French" phone shown as they talk about the intrusive unwanted guest calling Tom wasn't invented until about the mid-30's. In 1922, you still had to hold the candlestick phone in one hand and talk into the mouthpiece while holding the earpiece up to your ear.
Chosen answer: The song is "Back to Black" by Beyoncé (Feat. Andre 3000).