Symphonies 1 and 15, Lopez-Cobos, Cincinnati SO, Telarc
Symphony No. 5, Bernstein, NY Philharmonic, Sony (I prefer the earlier recording)
Symphony No. 7, Bernstein (I like both the NY and Chicago recordings)
Symphony No. 8, Haitink, Concertgebouw, Decca
Symphony No. 10, Ormandy, Philadelphia O, Sony
Violin Concerto No. 1, Vengerov, Rostropovich, Teldec (or Oistrakh and Mitropoulos)
String Quartets, Fitzwilliam Qt, Decca
The Lopez-Cobos and Haitik recordings have especially good sound, all of the performances are very fine.
I don't know about the "best," but I would recommend a recording of the Cello Concerti, Nos. 1 & 2, Truls Mork, Cello, and Mariss Jansons leading the London Philharmonic. Virgin Classics, VC 5 45145 2.
Mork lacks the big sound that Rostropovich offers, but this recording is gorgeously recorded with a wonderful balance between soloist and orchestra. The cello sound is really spectacular. I've given it as a gift because I wanted to share the experience.
sorry to get in late on this, but since no one else has mentioned it, i should add symphony no.13 "babi yar," with kurt masur and the ny philharmonic. it's historically important: the symphony is based on a poem written by yevgeniy yevtushenko, who was permitted, during the krushchev era, to say many things not permitted earlier about russia and the soviet union. "babi yar" is about a mass grave of jews and other minorities in russia, part of the wwii and stalinist legacy. this particular recording begins with yevtushenko himself reciting the poem first. the symphony is an interweaving of the sung words of the poem and the orchestral music shostakovich created around the meaning of those words.
I can also recommend the Borodin Quartet's later performances of 2, 3, 7, 8 and 12 on Virgin 61630 (upc 72435616302). It is a 2-cd digital set in the same vein as their earlier Melodiya/RCA/Chandos performances but for only $12.
big ditto on the string quartets...Borodin or Emerson (I have both complete sets). Let's not forget Symphony #4 either. That one was held back for many years due to the whole Soviet political thing. It certainly presents itself as dark, stormy & turbulent. Truly excellent.
I recommend the early version of the late quartets by the Borodin on EMI/Melodiya. I also recommend the cd from Columbia Heritage (I believe) that has the 1st Violin and 1st Cello Concerto with Oistrakh, Rostropovich and Eugene Ormandy conducting. For a very different side of Shostakovich (the pieces he "wrote for his desk", try his preludes and fugues. I have an old Russian record of him playing them himself, which is amazing, but Tatyana Nikolaeva gives an excellent rendition as well. Keith Jarrett is also enjoyable to listen to, but a bit different. For symphony's, I am partial to Eugene Ormandy. He has a double CD of the 4th and 10th symphonies which is very enjoyable and well recorded.
Edit: If anyone is insanely into Shostakovich, let me know. I wrote my undergraduate thesis on him. So if you want to read 80 pages of pontification on how he used music to express his views on the soviets, or you just want an extensive bibliography for further reading on him, I'm your man.
Many new recordings continue to surface recently........as far as reliable symphony performnces that have stood the test of time you can safely purchase Jarvi/Chandos for Symphonies 4-10 consistently very good performances and Chandos sound. Jarvi has no lack of energy when called for and holds the sometimes chaotic pieces together wonderfully, the downside is they are still full price.
Some other must have classic performances include Karajan/DG 10th, and Bernstein/DG/CSO 7th. Rattle and especially Jansons have recently released well received performances, I have been buying some of the Jansons series, he learned under Mravinsky
I would start with symphonies 5,8,10 and then expand from there.
But my favorite Shostakovich Cd is the Jarvi/Chandos 2CD collection of dance suites, amazing stuff that will really surprise most people since you don't associate him with this style: Jarvi
The most absolutely stirring recording of Shostakovich was one that I heard on the radio of a live performance of the 9th Symphony with Leopold Skokowski & the Houston orchestra - I tuned in in the first movement and found myself unable to do anything but listen, and then i could not believe who it really was. Not bad fidelity, as I remember, albeit on FM.