You will have to ultimately be more specific for a decent answer. The Soviet Union stretched from 1918-1991 which means tastes changed over the duration. It would be the same as asking what was America's favorite movies. There is simply too many to cover.
The urban legend goes that street crime fell during the time they showed Seventeen moments of spring, as everybody was inside, watching the show.
One tv movie series (sort of along the lines of Sherlock) was a late 70s/early 80s brilliant adaptation of Sherlock Holmes. A lot of it's on YouTube now. Really revolutionized the Holmes style, because it really updated the characters to a more modern style. Probably the first time we get a really competent and able to fight Watson. Rumor has it that the Brits couldn't let the Russians get away with having the best Holmes adaptation, so that's how the Grenada series was made. Even then, the soviet version is still considered the best by many Holmesians. If you watch the first, power through the weird park scene where Watson and his friend (the man who introduces the two) have this completely random conversation about how it sucks that people just can't have a private conversation in public without worrying who's listening in. I'd probably detract it, but whoever can work around the Soviet censors successfully in a made for tv movie gets a pass.
But seriously fantastic adaptation. Still better than RDJ, Sherlock, and Elementary. Especially Sherlock.
Hello! Let me try to answer, though I'm not a historian.
In USSR(at least after 1940) the success of a film was measured by number of tickets sold during the first year. Unfortunately the data were classified in Soviet times and even now they are not freely available. What we have - some data collected by researchers. One of them is Sergey Kudryavtsev (link in Russian), Russian film critic, who worked in different cinema-related companies and taught at Gerasimov Institute of Cinematography, the main film school in Russia. He published his data in books. Unfortunately the book is very hard to find, but he piblished some of the data on his blog(here: http://kinanet.livejournal.com/14172.html , in Russian). So, here are the data from his blog:
Top 10 Soviet films:
"Pirates of the XX century" (Gorky studio, 1980, action, adventure, by Boris Durov) 87,6 million viewers with the circulation of 1244 copies, and the highest number of viewers per copy - 70,4 thousand.
"Moscow does not believe in tears" (Mosfilm, 1980, social romance-melodrama, Vladimir Menshov) 84,4 million viewers, 1910 copies.
"The Diamond Arm" (Mosfilm, 1969, eccentric Comedy, Leonid Gaidai) 76.7 million viewers, 2079 copies.
"Kidnapping, Caucasian Style" (Mosfilm, 1967, eccentric Comedy, Leonid Gaidai) 76,5 million viwers, 1474 copies.
"The wedding in Malinovka" (Lenfilm, 1967, historical-revolutionary eccentric Comedy, Andrei Tutyshkin) 74,6 million viwers, 1815 copies.
"The crew" (Mosfilm, 1980, cine novel of Alexander Mitta with elements of disaster film), 71.1 million viewers, copies 1556.
"Operation "Y" and other adventures of Shurik (Mosfilm, 1965, eccentric Comedy, Leonid Gaidai), 69.6 million, copies 1197.
"Shield and sword" (Mosfilm, 1968, war the heroic adventure film by Vladimir Basov), 1 epsode "Without the right to be yourself" - 68.3 million, 2 episode "Ordered to survive" - 66.3 million, circulation - 1974 copies.
"The new adventures of the elusive" (Mosfilm, 1969, historical-revolutionary eccentric movie Edmond Keosayan), 66.2 million with circulation 1920 copies.
"But the dawns are quiet here" (Gorky studio, 1973, a military-heroic cine novel by Stanislav Rostotsky with elements of melodrama), 66,0 million viewers, copies 1717
In his blog post and comments you can see that this list is apparently incomplete and some numbers are questionable. He also posted list of top ten foreign films in USSR, but I will post it later.
This is one of my favorite Soviet films, and it was enormously popular at the time: http://en.wikipedia.org/wiki/Letter_Never_Sent_(film)
I have a side question: were there any American shows that were popular during the Soviet Union, specifically during the Cold War Era?