I heard Ivan Rebroff once, but I am better now
If you want to hear someone from the current generation who does this kind of repertoire, there are three decent basses you might look out for... Herman Yukavsky sings a lot of the "popular" Russian songs, and sometimes even accompanies himself whilst doing so (I'm not a great fan of this stuff, but his performance of "Ulitsa", as a drunk returning home in the early hours, is side-splitting stuff). Dmitry Schwed (who could be heard as Banquo in the Glyndebourne Prom of Verdi's MACBETH recently) does all the popular ones and some of the "classical" songs by Mussorgsky, Tchaikovsky, Taneev etc too. Dmitry Skorikov is a talented young bass who mostly does the more "classical" songs (Rachmaninov, Prokofiev)... a multi-talented guy who can be found singing both Pimen in BORIS GODUNOV, and "Mack the Knife" (complete with a tap-dance routine).