Well, I am not familiar with these enough, just because they aren't popular enough and not easy to get. However, rather many people praise "Юнона и Авось", "Норд-ост". I know only one song from the former - something like "Я тебя никогда не забуду", if I remember the words properly - it's a real hit, often on radio (even though it's really an oldie, as well as the whole musical), really touching melody... It must be on Youtube, I guess...
However, originally Russian musicals are really rare. And... I am much more familiar with Russian operas (I adore "Eugene Onegin", know the libretto practically by heart; then "Pique Dame" is very nice, and Borodin's "Prince Igor", and Musorgsky's "Boris Godunov"...)