This question is incredibly broad, so it’s going to be difficult to answer thoroughly. I also don't have quite as much time to write as thorough answer as I'd like, but I also don't want to let a question like this go unanswered, so I'll take a stab at it:

The relationship between the state and the arts in the Soviet Union changed dramatically over time from the Revolution onwards, with the harshest repression coming, as one would probably expect, under Stalin. I recently wrote a post about the arts in the Revolution (and 1920s) that you can find here. In your follow up comment you mention the 1950s and 60s in particular, so I am going to answer the question about the arts under Khrushchev in particular in order to have a workable timeframe.

When Khrushchev came to power, the Soviet Union faced a pretty major shift away from Stalin and Stalinism. In deliberately moving away from Stalinism, Khrushchev kicked off a period that quickly became known as “The Thaw.” Although this was a period of reform, it shouldn’t be understood to have completely overturned cultural repression by any means. On the one hand, artists did have considerably more ability to cover topics that would have been censored under Stalin. On the other hand, Khrushchev still believed in the importance of the arts and culture to socialism. In other words, artists had more freedom, but that needs to be tempered by the fact that the bar was quite low to begin with. Khrushchev also opened up the Soviet Union to outside culture much more than had been possible under Stalin. For example, we saw international film festivals inside the Soviet Union. Nonetheless, when Pasternak was given the Nobel Prize for literature in 1958, he had to renounce the award. Sound contradictory? Welcome to studying the Soviet Union!

Pasternak’s example here gets more to the heart of your question. If you were an artist in the Soviet Union you had to navigate censors, you had to consider what opinions might be rejected, and you did have to tread carefully, even under Khrushchev – who took a very personal role in commenting on the arts. (http://soviethistory.msu.edu/1961-2/khrushchev-on-the-arts/) It’s also important to point out that censorship was a process, not a single action taken by the government (usually). Works were presented to censors, revisions were made and this process could go back and forth many times before a work was deemed acceptable. This, of course, applies to art being produced through official channels and by members of artists unions. It was a rather unstable time to be an artist, and it was difficult to know what would or wouldn’t be accepted.

That’s to say nothing of those producing art outside of official channels. The 1960s saw the rise of bards in the Soviet Union. The term didn’t refer to a particular organization, but rather musicians and poets who were producing work outside of the official system. Their work was not even always explicitly hostile to Soviet rule, but many songs were. The nature of the work meant that they played small audiences and couldn’t really advertise their shows in any meaningful way. Some even became rather famous within the Soviet Union, like Vladimir Vysotsky. Vysotsky became a film actor and was well known for his social and political commentary, which was mostly tolerated. Much of his fame came under Brezhnev, rather than Khrushchev, just to be clear. Vysotsky’s work is all over youtube, for example: https://www.youtube.com/watch?v=OBLWDk8q6Tc

Recommending Reading:

Fitzpatrick, Sheila, The Cultural Front: Power and Culture in Revolutionary Russia (1992) - See also the other post I wrote and linked to up at the top of the page, I mention several other works to do with the early Soviet Union.

Hosking, Geoffrey, Beyond Socialist Realism: Soviet Fiction since Ivan Denisovich (1980)

Jones, Polly, ed., The Dilemmas of De-Stalinization: Negotiating Cultural and Social Change in the Khrushchev Era (2006)

Stites, Richard, Russian Popular Culture: Entertainment and Society since 1900 (1992)

Caute, David. The Dancer Defects: The Struggle for Cultural Supremacy during the Cold War (2003) - if you're interested in ballet/dance in the Cold War.