In my experience Western music had a profound influence on youth during the Soviet period and spawned movies about Russian counterculture such as the fantastic and lively Stilyagi.
Someone once gave me a list of music groups that were banned during Soviet times (that I unfortunately no longer have). I'm sure the list only made people want to hear the groups even more, which is usually what happens with such things. Anyway, with a quick Google search this is what I found:
“Group Name and Type of Propaganda”
1. Sex Pistols - punk, violence
2. B-52s - punk, violence
3. Madness - punk, violence
4. Clash - punk, violence
5. Stranglers - punk, violence
6. Kiss - neofascism, punk, violence
7. Krokus - violence, cult of strong personality
8. Styx - violence, vandalism
9. Iron Maiden - violence, religious obscurantism
10. Judas Priest - anticommunism, racism
11. AC/DC - neofascism, violence
12. Sparks - neofascism, racism
13. Black Sabbath - violence, religious obscurantism
14. Alice Cooper - violence, vandalism
15. Nazareth - violence, religious mysticism
16. Scorpions - violence
17. Genghis Khan - anticommunism, nationalism
18. UFO - violence
19. Pink Floyd - distortion of Soviet foreign policy ("Soviet aggression in Afghanistan")
20. Talking Heads - myth of the Soviet military threat
21. Perron - eroticism
22. Bohannon - eroticism
23. Originals - sex
24. Donna Summer - eroticism
25. Tina Turner - sex
26. Junior English - sex
27. Canned Heat - homosexuality
28. Munich Machine - eroticism
29. Ramones - punk
30. Van Halen - anti-Soviet propaganda
31. Julio Iglesias - neofascism
32. Yazoo - punk, violence
33. Depeche Mode - punk, violence
34. Village People - violence
35. 10cc - neofascism
36. Stooges - violence
37. Boys - punk, violence
38. Blondie - punk, violence