In the bureau of housing Ostap Bender, disguised as a soviet official, is writing with his left hand. However in the Soviet union, especially in the time the movie is set, left hand writing was not tolerated by the state and would be hardly accepted in public, as Bender is writing not only hidden behind the shelfs but was also about to write in front of Father Fyodor. Left hand writing was accepted in USSR only in 1985-1986.
Shortly after Ippolit and Ostap arrive in Moscow, there is a pullout shot of Soviet buildings with prominently-displayed television antennas atop most of the buildings. Television broadcasting did not start in the Soviet Union until 1938--11 years after the date of this movie's setting (1927).
Russian railroad tracks are famously unique for being laid with their joints across from each other rather than being staggered as they are elsewhere. When Ippolit and Ostap walk walk back to Moscow all the tracks they pass have staggered joints.
On the car of the Columbus Theater for the announced Play "Hamlet and the October Revolution" next to Shakespeare the name Ivan Poppov is mentioned, obviously a reference to Ivan Ivanovich Popov, a Russian revolutionizer, journalist and freedom fighter. However he is written with one P, not two.
The title of the movie is displayed in Russian (Cyrillic) before it transitions to English. If you look closely however, the Russian spelling for the movie's title is incorrect. The title is misspelled as the Cyrillic equivalent of "Dvenadtsat Stchlev" instead of the proper spelling, "Dvenadtsat Stulev".