In terms of Russian singers, I recommend checking out:

Boris Grebenshikov  (Борис Гребенщиков) https://www.youtube.com/watch?v=o0y-yLcdp3s

Viktor Tsoj (Виктор Цой) https://www.youtube.com/watch?v=FU8csnZxdPA

Yuriy Shevchuk (Юрий Шевчук) https://www.youtube.com/watch?v=V4Ui6iKAU_o

Eduard Khil: https://www.youtube.com/watch?v=5jfWmdDZq1A

                  https://www.youtube.com/watch?v=aGVtcfx2TU4

Pelageya (Пелагея). https://www.youtube.com/watch?v=FaRcatKStkM

https://www.youtube.com/watch?v=6oPK4syeSGc

https://www.youtube.com/watch?v=bvAQvDvyvxk

Oleg Medvedev: https://www.youtube.com/watch?v=3i-6VdOQXTY

Би-2. https://www.youtube.com/watch?v=2FW68e2668w

There are many more good bands and singers, but the ones listed here are a good start. 

I like the early russian rock music. The eighties was the legendary time for Russian Rock. It was the time when the greatest russian rock artists began thei carreers. They are: Boris Grebenshikov ("Aquarium" band), Victor Tsoi ("KINO" band), Yuri Shevchuk ("DDT" band), Konstantin Kinchev ("Alisa" band). They are the greatest. There are also a lot of good legendary bands - "Mashina vremeni" (Time machine) with Andrei Makarevich, "Chizh i kompaniya" ("siskin and company", siskin - sounds fun in english ;). Check the wikipedia for the Russian Rock Club.

Of the modern russian artists i prefer to listen to Zemfira, Luna, Nochnie snaiperi (Night snipers) and others. I can show you the easy way to listen to russian music. Ask me in private message if you wish. :)

Listen to Zemfira: http://www.youtube.com/watch?v=xpJ_fYyAeYk

Dima Bilan: http://www.youtube.com/watch?v=JALDq9hagKQ

When I was younger I used to listen to bands such "Dolphin", "Fleur", "Маша и Медведи" (Masha and Bears), "Агата Кристи" (Agatha Christie) and others which were already mentioned. Modern music makes me sad :(

Time Mashine- Машина времени

Rock group)))