There was no such thing as an underground standup scene in the USSR (with one small exception, I'll talk about that later).

I'm not sure we can call this stand-up but there were artists who told funny stories on stage in the USSR. Let's call them standupers, to simplify things. Also, I'm talking about 1970-1980. To some degree, it's relevant to 1960s also.

There were two different categories. The first one is authors who read their own stories. The most famous were Mikhail Zhvanetsky (this guy was really really good), Mikhail Zadornov (very funny in his peak, slightly crazy after that) and a lot of good writers lesser caliber (Altov, Gorin, Arkanov, to name a few).

In russian they were called писатели-сатирики (satiric writers).

Actually, Gorin was a very talented author, like first-class writer, but he's most famous not for his funny stories, but for screenplays and plays. They're very, very good. I don't know if there are Формула любви (Formula of Love) or Убить дракона (To Kill a Dragon) or Тот самый Мюнхаузен (The Very Same Munchhausen) with english subtitles, but these are one of the best Soviet movies of all time, and they're really really relevant today, especially To Kill A Dragon.

But back to satiric writers. Sorry for that. These writers read their own stories on stage. Let me repeat it: they literally read them, with papers in their hands, nobody cared to remember anything. And this format was quite popular because of the quality of writing and, maybe, the lack of other forms of entertainment.

People could access this content on TV (there was a very popular TV Show Вокруг смеха) or they could buy pirated tapes or they could attend some live shows in their cities.

The second category is actors. These guys could remember their texts. And mostly these texts were written by some guys from the first category. So, it's not surprising that show Вокруг смеха (I mentioned it earlier) consisted of writers and actors, roughly 50/50. Roughly speaking, there was three groups of people: 'silent' authors who sold their stories to actors, actors and writers who sold their stories to actors and also read some unsold stories by themselves.

The reason for existence of the last group, I believe, is money. You could earn much more if you read your own stories than by selling them to other people.

Most famous actors were Arkadiy Raykin, Roman Kartsev, Gennady Hazanov, Evgeniy Petrosyan (I, personally, not a fan of Petrosyan, but he was really popular). Raykin was kind of a patriarch there. A lot of talented writers and actors began their career in his own theater (not really his own, it was in the USSR, after all, but you get the idea).

It wasn't really underground. All these text had to be approved by special commissions. All these guys — writers and actors — weren't able to tour by themselves, they had to belong to some concert organization to do that. So, it was quite strict. And therefore there was no improv, actually. The first elements of improve appeared after Perestroyka and Glasnost (1986-1987). I believe the first comedians who tried to use elements of improvisation in their shows were Yan Arlazorov and Mikhail Zadornov. But still it was just elements, the main part was heavily scripted.

I promised to tell you about some exception. Here it is. Although it was very hard to say something against authorities on TV or on stage, there were also small unofficial shows (in some small provincial theater or even in someone's flat). The main reason behind these shows was money, but still authors often allowed themselves to be more brave and critical and funny. Pirated audiotapes of these shows could be later copied or bought. Of course, it wasn't very hard critique and it wasn't extremely anti-Soviet (far, far from it), but still.

I already said that Soviet format wasn't too similar to modern standup, but there is one more difference. Because they couldn't say a lot of things, they sometimes used the power of silence and absurd. Like, you are saying several quite innocent sentenes and then you are just waiting when your listeners complete the picture in their minds. The best jokes weren't pronounced at all, they existed only in imaginations.

Modern standup shows — even the best ones — are much more interactive and direct. I don't want to say that they're not so good, but Soviet 'standup' was much more nuanced and sometimes provided this strange but warm feeling of shared understanding, when people around you were laughing on the same joke as you, but noone actually had said this joke out loud.

The best examples of Soviet comedian writing is more like a poetry. It's a beautiful and somehow undervalued part of the Soviet literature.

Oh, I'm sorry, I didn't really answer your question. I kind of answered because the answer is in my reply, but there are a lof of other words and thoughts, sorry for that.

TL/DR: If you want to be a part of standup scene in the USSR in 1960-1980, there are two ways. First, you can try to become a writer. For this you have to publish your stories in newspapers (Литературная газета is your best bet, because it has a whole page just for funny stories), magazines (Крокодил) and after that you really want to become a stock writer for some satire theater (Arkady Raykin's theater is good, Satire Theater in Moscow is also good). If you'd be able to become a resident of TV Show 'Вокруг смеха', and you're good, that's it. You're famous and loved.

For actors it was almost the same. Raykin, Satire Theater, some Odessa theaters (Odessa was part of Ukraine, and Ukraine was part of the USSR at the time), and after that TV and movies. Congrats, you're famous and loved.