You won't find the equivalent of John Wayne because Red Westerns featured Native Indians as the heroes, often played by Gojko Mitić.

Perhaps not Bugs Bunny but more of Wile E. Coyote and The Road Runner is Nu, pogodi! a cartoon about a Wolf and a Hare.

Soviet Sinatra was Joseph Kobzon. More diverse and entertaining was Muslim Magomaev. Soviet Beatles was the band called Secret. Polish singer Anna German was very popular.
How it was different from western pop-culture? They had to sing about good things. No YMCA or Mothers little helper. Music had to touch positive strings in peoples souls: love your country, love your village, love your family, be ready to accomplish something big. Either that, or no LP and no TV translations, because The State owned them.

About Elvis - I'm not sure there was one at the time but Joseph Kobson, Muslim Magomaev and several others were extremely popular in the 1960s. But I have to say that celeb culture per se was not encouraged. So it was ok to buy LPs but it was not very ok to cry and scream at live concerts, this type of behavior was classified as "bad influence from the west" (тлетворное влияние запада). The nearest equivalent to Kobson, Magomaev and especially Mark Bernes and Leonid Utesov is Frank Sinatra.

1970-1980s. At this time we have several real rock heroes. The first one is Andrey Makarevich and his band 'machina vremeni' (time machine). This band was extremely popular and they were lightly supported by authorities - there were official LPs and sometimes (very rare) you could watch them on tv (1980s). It's kinda hard to find Russian who had never heard of them. Their music is very melodical and quite easy to absorb (and slightly annoying, imo) - but they created a lot of catchy tunes.

The second one is Aquarium and his leader Boris Grebenschikov. They were much more underground but they were popular amongst university students and they had at least one super hit - the golden town (город золотой). Boris is kinda Russian Bob Dylan and Aquarium is like Russian Jethro Tull. Boris recorded one or two albums in English, so it is quiet easy to understand what his music is.

The third one - is the real Russian Elvis and the Russian Kurt Cobaine as well. And he is Korean. From the 1982 till 1990 band Kino and its leader Victor Tsoi were the main Russian musicians for anyone age 12-25. Victor is the legend, and his death in 1990 just made this legend stronger. I think their music was similar to the first albums of the Cure.

Also I should mention that all of these guys wrote very very good lyrics. Russian rock music was not about music only, and maybe it was not about music at all, but about lyrics.

Bugs Bunnie - I don't think there was something like that but we had cartoon series "hey, wait" (ну погоди) that was similar to Tom and jerry series, but instead of cat and mouse you have wolf and rabbit. Every Russian kid knows these cartoons.

I want to add that there is also absolutely crazy and funny Russian version of Winnie the Pooh, very funny series about uncle Fyodor and very good series about Swedish Carlsson (also quite different from the source).

John Wayne - it's a hard question because we had a lot of popular movie stars. It's only my opinion but I think that beauty standards for male stars were slightly lower. For example, one of the most popular Russian actors (the one whose voice was behind Winnie the Pooh) Alexei Leonov was like Denny de Vito; another one - Leonid Kuravlev - had quite simple and open face. The talent was much more important than the looks. But there were sex symbols as well - I think Michael Boyarsky and Andrey Mironov were them (the second one was very talented guy).

It might help if you specified a country and a time.

If the question is whether communist countries had a distinct popular culture, the answer is definitely yes. I'm not sure how you would go about measuring its strength, and its different aspects are regarded differently these days, but yes, there was an Eastern bloc pop culture.

There were a lot of cartoons produced in Czechoslovakia during the communist era. The program Večerníček, which is still running, showed cartoons like Krtek, Pat a Mat, and Maxipes Fík. They're pretty fun and still very much beloved in the Czech Republic and Slovakia. Karel Gott and Helena Vondráčková are examples of pop music stars from the 70s and 80s -- both are still around. Vondráčková was in a group with a woman named Marta Kubišová, who was also a singer but expressed dissident opinions and was banned from performing in the 1970s. Oh, and Michal David is kind of a symbol of bad music from the 1980s. The television show Thirty Cases of Major Zeman is a good example of obvious propaganda -- the protagonist is a police investigator, and many of the episodes were based on real events, with a pro-government sort of spin.

East Germany had the Puhdys.

Thanks everybody!

In the sixties an American singer named Dean Reed moved to East Germany after becoming politically radicalized in Latin America (where his records were extremely popular, even while Reed remained completely unknown at home).

Through the seventies and early eighties Reed became very popular throughout the Eastern Bloc by performing covers of classic rock songs and his own politically charged originals. Many even called him "the Red Elvis."

As his popularity waned in the eighties - and following a disastrous appearance on 60 Minutes, during which he defended the Berlin Wall - he took his own life. (Not surprisingly, many still believe the CIA/KGB/Stasi had something to do with it.)