I'll chime in another recommendation for the Shostakovich Quartet. Their mellowness is a little different from what one typically hears today, but they highlight the music's debt to the Russian Romantics of the pre-Soviet era. For me, nobody betters the Shostakovich Quartet in their sweetly nostalgic performances of the Sixth and Fourteenth Quartets. Their recording of the Fifteenth has a more consolatory quality than usual; a representation of death as friend, rather than foe to be feared. In general, theirs is an almost Schubertian Shostakovich.
The Beethoven Quartet is a must-hear. They premiered all but Quartets 1 and 15 (founding member Sergei Shirinsky died in mid-rehearsal of the latter) and collaborated with the composer closely. Their sound is wiry, narrow, heightening the nervous tension in Shostakovich's quartets, but with occasional surprising portamenti that imparts a sense of vulnerability unlike anything else in the discography. (Try their earlier recording of Quartet 3, last available on a Symposium CD.) To my ears, no other quartet better renders the humanity of this music than the Beethovens. The joy, sarcasm, and bitter anger of this music all come through, sitting cheek-by-jowl, with a directness and comprehensiveness that no other quartet matches.
On the other hand, the Borodins' take on the composer is unrelievedly grim, granitic, imposing. As much as I like their earlier sets, their Decca set, I thought, was excellent, perhaps their best. It has an airiness, a willingness to let the music breathe that at least for me is convincving. (Of course, their earlier performance of the Piano Quintet with Richter is a milestone of the catalog that demands to be heard. There was also a live UK performance on Intaglio that, perhaps, may top the better-known Melodiya recording.)
Intimacy and humanity mark the Fitzwilliam's cycle. Warm, rich, full-bodied performances that contextualize the music as successors to Beethoven. The birdsong inadvertently captured in some of their recordings (try the coda of Quartet 3) is one of the great fortunate circumstances of recorded music. Especially revelatory are their performances of the late quartets. (Shostakovich coached them in Quartet 13.) Rather than being the documents of desperation one usually hears, the Fitzwilliams find a sense of hope, however slim, in this crepuscular music.
Of modern cycles, my favorite is the Mandelring on Audite. Full-bodied, deep, oaken in tone; theirs is Shostakovich as successor to Brahms. The Audite production is gorgeous. A lot of people like to hear Shostakovich as angry hysteric, but I also enjoy very much hearing him as part of the Western musical tradition. (Which is why I love East German recordings of his music.) Shostakovich was not only a rousing, angry denouncer of his own time and place, but also an artist who was deeply indebted to Austro-German traditions, and who was proud of his place within it.
Also, yes, the Pacifica are wonderful; especially for the historical context provided by their imaginative pairings.