I think it depends. Everyone knows about them and the tunes are famous. What about lyrics I doubt many people know them by heart, especially young people. Most can sing along to refrain or remember a few lines, but not the whole song. These are songs I can imagine my Grandma singing. My mom? I don't think so. )
PS. Speaking about song choice for a Russian language learner I can't help mentioning (again) "KINO" songs.Very clear pronunciation, relatively slow tempo and relatively easy grammar constructions, modern vocabulary, very little or no slang.
http://www.youtube.com/watch?v=1HKXMBfDxR0
http://www.youtube.com/watch?v=sMT9F7dZltY