This is surprisingly a very interesting subject to delve into, and you're in luck, Chuck because I happen to know quite a bit about it. Some of you may already have an image in your heads of a peppy proletarian wearing farmer's clothing with a Budenovka atop his head, maybe a wrench in his hand. Maybe you're thinking of a shivering soviet wearing a thick trench-coat and a ushanka on his head. Both of these aren't entirely accurate when it comes to the USSR as a whole.
It really depends on what decade of the Soviet Union we're talking, it also heavily depended on your area. To make things simple, I'll talk about the Soviet Union as a whole.
Let's start with the 1920s, before the infamous Man of Steel, also known as Joseph Stalin (Not Superman though, at least not this timeline)
There was definitely some Capitalist fashion leftover. There were still magazines and advertisements. Except there were many who opposed Western fashion, they felt it was just another facet of the bourgeoisie and also pretty sexist towards women. There were magazines that promoted more gender neutral clothing that was more practical.
Once Stalin took over, he decided that the fashion of the 1920s encouraged the myth that Communism had to be uniform and drab. State-encouraged fashion was extremely extravagant and unique. This actually made a lot of people pretty upset because these fashions were out of their grasp by a mile. There were issues with producing these fashions and often times they wouldn't be a priority.
The solution to this problem was to publish magazines with tutorials for Soviet women to make their own clothing using whatever they could find lying around. It just wasn't the same though and this would heavily influence future Soviet fashion.
Once we get to the Khrushchev Thaw, the fashion pendulum (picture a regular pendulum, except bedazzled) swung in the complete other direction as a way to compensate for the backlash from Stalin's attempts at fashion. Official clothing became more like the West, except a lot more plain, modest, and traditional. These were somewhat more accessible, but not as bold or original. There were those in the US during this time who accused the Soviets of resorting to stealing Western fashion.
This is actually when a sort of subculture started to arise in the USSR, they were known facetiously as, "Style hunters." During Stalin's reign, they were violently repressed by those who I hope referred to themselves as the, "Fashion police" when they burst through their fellow comrade's doors, but the Thaw allowed them to really shine. These were people who wore stereotypically Western fashions, often times very shallow impressions gleaned from American movies that were shown in the Soviet Union. These were people who were pretty discontent with the status quo, and this was their way of rebelling.
Now when we come to the 1960s, the rule of Brezhnev, we see yet even more changes. The state let out a collective sigh and basically embraced Western fashion a lot more. Secondhand shops popped up to buy used clothing off tourists to sell them to eager Soviets.
However, the state also produced sort of, "Space Age" fashion in response to the Space Race. These included clothing based on cosmonaut suits and Soviet futurism. It's exactly as dorky as it sounds, I assure you that.
Now when we get to Gorbachev and Perestroika, we see even more openness towards Western fashion. Less media was censored from the West, exposing the average Soviet to all the sweet fashion they were missing. Of course, the state did attempt to meet demands by producing more consumer goods, but it had quite a few snags along the way. It was still pretty hard for the regular Joe to get his hands on brand new fashion.
That pretty much summarizes it, feel free to ask me any more questions you may have.