In the radio booth, at around 4 minutes, Andreyev is still debating the correct time to return Stalin's call with Sergei, who is seated with his headphones around his neck. But, in the shots with the camera over Sergei's shoulder his headphones vanish, only to reappear in the next front-on shot. After several cuts like this, Sergei's headphones finally disappear altogether.
When Lavrenti Beria (Simon Russell Beale) is arrested and led off bound and gagged by a couple of Red Army soldiers, his face has no blood on it, while prior and subsequent scenes show his face beaten and bloody.
When Stalin's son tussles with security guards his air force badge is knocked sideways, immediately as he stands up it is in the proper position.
Malenkov did not become General Secretary of the Communist Party when Stalin died. He did, however, become Premier of the Soviet Union. The Soviet leadership was clearly in flux, and Malenkov never had the status of obvious successor to Stalin that the movie implies. He did not chair the Politburo meeting after Stalin died; Khrushchev did.
Zhukov is introduced with the caption "Field Marshal Zhukov - Head of the Soviet Army" The rank of Field Marshal was abolished in Russia in 1917, Zhukov's rank was Marshal of the Soviet Union (he is usually referred to as Marshal Zhukov). Nor was he head of the army at the time of Stalin's death, as he had fallen out of favour with Stalin.
Svetlana asks Beria to release Aleksei Kapler, her first love who is imprisoned in the gulag. Beria tells her that Kapler is dead. However, Kapler was released in 1953, soon after Stalin's death, and lived to 1979. There doesn't seem to be any reason for Beria to lie, since he is supposed to be ingratiating himself to Svetlana, and as the head of the NKVD he certainly would have known that Kapler was still alive.
Zhukov is referred to as "Generalissimo." This is in fact a rank one degree higher than his actual rank of Marshal (six star as opposed to five star in American terms); it was proposed, for Stalin only, but never implemented.
Vyacheslav Molotov in real life was known for his impassive, stern demeanor, while Michael Palin depicts him as emotional, somewhat cowardly, and seeming very pleasant and cheerful.
Several times, at 58:17, 1:31:27, and 1:35:40, the semi-auto TT-33 pistols used don't cycle when fired and no spent casings are ejected from it. The sound-effects and muzzle flashes are added in post-production.
At 59:20 when Kruschev is seen debating the curtain in the background the dead body of Stalin is breathing (his chest moves).
By the time of the Stalin's death, the NKVD was non-existent: it was reorganized and became MVD in 1946. It was again modified March 15, 1953.
Stalin died on March 5, 1953; however, the movie shows the trees in full bloom. The trees would not have had any leaves in bloom that early in the year.
At 4:40 the cars roll to Moscow down a freeway with modern corrugated steel crash barriers. Such barriers were only invented in 1970s and were not in use in Russia until 1980s. Photos from that period show Moscow area highways without any pedestrian protection.
The film implies that the plane crash that killed many of the Russian Air Force's hockey team that Vasily coached was recent. However, it happened in January 1950, three years before Stalin's death.
In the concert scene, at least one cello appears to be using a modern combination of Larsen A and D and Thomastik Spirocore G and C strings. Larsen strings were not manufactured until about the 1980s and it is doubtful that Thomastik strings (of Viennese origin) would have been available in Soviet Russia.
Faced with Stalin's imminent death, Svetlana says that she knows "doctors in Stalingrad". Stalingrad is an industrial city far away in Southern Russia. Surely, the writers meant Leningrad (St Petersburg) which would indeed have many distinguished doctors. Svetlana doesn't seem to have spent any time in Stalingrad, so there's no reason to think that she would have known doctors there.