Unlike his 15 symphonies, it's hard for me to single out a few as my favorite quartets. Maturity may be part of the reason, as they all came after Symphony #5, but not all, since Symphony #1 is a youthful product and one of my favorite too! Some experts explain and contrast this set as related to his private persona vs the 15 symphonies to his public persona. Again, there exceptions to the rules, where Symphony #14 is more like his late quartets in feeling if not in scope, and Symphonies #6, #9 and #15 feels somewhat like his more lyric and fun-loving quartets.
These 15 quartets are my personal favorites among all the quartet literature, truer and dearer to my heart than any other quartets. To be honest, I had lukewarm feelings towards quite a few of Beethoven's quartets, and consider only LvB's later quartet to be on par with Shostakovich's. I know that Bartok's 6 quartets are highly praised, but I still have a hard time to listen to them for long or for frequent listening. Maybe it's just not a good match to my "personality" type.
Borodin Quartet did wonders to these quartets. I can still remember, vividly, as if it's just yesterday, the group's performance of the last three quartets in North York Performance Art Center near Toronto Nov. 8, 1994, one of most impressive performances I ever witnessed. I also heard Cleveland Quartet performing Shostakovich #8 at the University of Maryland UCC (University Community Concert series), truly amazing too. In fact, Cleveland Quartet is my other favorite group, where I also heard them playing Dvorak's "American" quartet and Ravel's only quartet, two of the works truly dear to my heart, besides Shostakovich's 15. It's a sad fact of life both these groups are no long exist in their original form. I have the last CD of Cleveland Quartet's farewell tour recording and will treasure it for a long time. And the reconstituted Borodin Quartet I heard in 2001 at Concertgebeuw, Amsterdam, simply lost the magic of the original group. A third group that I heard quite a few time, Juilliard Quartet, doesn't seem to be identified with Shostakovich much.
When it comes to the recordings, it's an easy choice for me. I'll take Borodin Quartet complete set on EMI any day! In fact, I have half of the set autographed after I attended their performance 11/4/1994 in Toronto. The partially completed cycle on Virgin is nice too, but unfortunately they didn't finish.
Two other sets I have listened to from time to time are by Fitzwilliam Quartet and Emerson Quartet. I compare Emerson set to that of Haitink set of symphonies, and Fitzwilliam set to Jarvi symphonies. If that's the analogy, then Borodin quartet set is like Rostropovich/NSO symphony set, truly personal favorites, but not to the exclusion of other interpretations. These works are simply too great to subject to only one interpretation.
I also have a few other versions, not complete sets. Several by St. Petersburg Quartet, nothing special to me despite the home-town link. Taneyev Quartet performance is nice, so is Yo-Yo Ma/Issac Stern/etc. on CBS Masterworks/Sony Classical.
Related to the quartet literature, there are also a few small pieces for string quartet by the master. The collection CD, "Russian Miniature", by Borodin Quartet on Teldec is an enjoyable CD that contain some such pieces and others, and that you can leave on for great background music.