Unemployment in the Soviet Union is difficult to track because the government claimed to 'liquidate' unemployment in the mid 1930's and didn't have reliable information or numbers. After the fall of the Soviet Union in 1991, many of the few records were lost so it's hard to give exact numbers, but there are some things that we do know.

1. Cyclical unemployment (unemployment resulting from economic ups and downs) was very small from the 1930's to 1945; after WWII unemployment started to sync with global recessions and booms and resemble western countries.
2. Frictional unemployment (unemployment between searching for jobs) follows a similar trend to cyclical unemployment, although it was notably high in the central Asian SSR's in the latter half of the 20th century.
3. Structural unemployment (unemployment caused by a lack/excess of skilled labor) was remarkably low during the Lenin era due to his educational policy and slower industrialization, but quickly rose after Stalin and his successors. By 1990 there was a crippling lack of non-industrial engineers and other jobs outside the production sector. The USSR had an especially harmful lack of doctors after the 1960's that couldn't keep up with the rising population.
4. Employment was generally higher in Russia and the East European SSR's and lower in the Caucus and Central Asian SSR's. By the 1950's the economy was globalized so even a relatively isolationist nation like the USSR was bound to be affected by global employment treads.

Basically, it can generally be said that the USSR did a fairly good job of employing people until after WWII when the war machine started ramping down. This isn't that that striking considering that is the pattern that most of the world took although it can be noted that the USSR did fare well during the 1929 depression and kept employment up.

Source:https://www.ucis.pitt.edu/nceeer/pre1998/1992-806-36-2-Moskoff.pdf (this is definitely worth a read, very interesting)

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Also, in USSR being unemployed was a criminal offense under article 209 of the criminal code. Joseph Brodsky was persecuted for it, for example.