A top member of the politburo visits a school. The teacher and the commissar stand in front of the kids with pictures of Lenin and Brezhnev.
The teacher asks the kids what country has the best food. The kids shout, “The Soviet Union!” What country has the most freedom? “The Soviet Union!” What country has the best toys! “The Soviet Union!” Then they see a kid crying in the back of the class. The teacher asks, “Mishka, why are you crying?" Mishka says, “I want to live in this Soviet Union.”
A man in the Soviet Union wants to fix his car so he visits a mechanic shop. They tell him the car will be ready and delivered in 8 years, on the 5th of June. He nods understandingly. He then asks when on the 5th of June it will be delivered. The mechanic is confused and asks why he cares. It will be many years until then, why bother with time of day? The man says, "Well, I have a plumber coming on the same day.”
1 of 2: A Russian and an American die. The devil asks them if they want to go to the capitalist hell or communist hell. They ask what's the difference. The devil says you have to eat one bucket of shit every morning in the capitalist hell and two buckets in the communist hell. The American chooses the capitalist hell, but the Russian chooses the communist hell.
(I’ve also found a few jokes making fun of specific Soviet figures or incidents, but I don’t know enough Soviet history to get most of them unfortunately. Some of them are still funny anyway.)
Like this one:
In the museum of Vasily Chapayev the guide shows the visitors a skeleton.
"Here you can see the skeleton of Vasily Chapayev."
"And what is this small skeleton next to him?"
"That's Vasily Chapayev in his childhood."