Recommendations please!
Vegh Quartet are superb. On LP: Telefunken 6.35040
I am happy with the Tokyo String Quartet.
Beethoven's late string quartets were composed between 1823 and 1826, and include Opus 127, 130-132, and 135. The versions of these string quartets that I prefer include:
1. Tokyo String Quartet (RCA 09026-60975-2);
2. Guarnieri String Quartet (RCA 60458-2-RG);
3. Quartetto Italiano (Philips 426050-2).

Good listening!!
Historic-budget choice The Cleveland Quartet

Contemporary The Emerson String Quartet,if you order on line be careful that you only order the last five;they recorded all of them.
For 15-16 I recommend the Talich Quartet (Calliope CAL 9639), for 12-16 + Grand Fugue the LaSalle Quartet (DG).
I agree with Rushton, the Vegh Quartet is superb. But then the Alban Berg Quartet on EMI is in a class of its own. They are absolute masters of ensemble playing and give especially 15 and 16 an intensity and passion, which is truly out of this world!
As the above recommendations

My Favorite: ABQ
My Wife's favorite: QI

Our Alternative choices: Tokyo and Vegh

Historical: Budapest
I agree, as I usually do, with Detlof. I have the Alban Berg live and studio versions of the quartets, plus parts of almost all the other quartets listed. The Alban Berg has long been one of my favorite groups for their extraordinary ensemble, and I always seem to resort to their Beethoven after listening to any of the others. I don't particularly care for the Emerson versions. The Tokyo give their best interpretations in the early Beethoven quartets, as opposed to the late ones.
I am surprised nobody mentioned Talich Quartet version? Performance and sound are exceptional
Data, Greg did mention the Talich, which indeed give a beautiful rendering. Does anybody, besides Plsl remember the Budapest Quartet? The old one, that is and the Borodin Quartet? They are high on my list as well, but like Flex, I also generally return to the ABQ, after listening to ANY other Beethoven interpretation, be it the early or late quartets.
While the ABQ has generally been my favorite quartet, I am quite taken by the Quatuor Mosaiques. After following the scores to their recordings, it's hard to go back.
Budapest SQ for a complete live set.
Hungarian SQ for a complete recorded set.
Capet and Busch for "historical".
Flex: interesting about liking the ABSQ but not the ESQ.
Is there a particular reason. The clarity and the muscular
presentation of ESQ are more in tune with what I thought Beethoven would have wanted. I am not saying I like one over the other; in fact I like the fluidness of the ABSQ's interpretation.
Busch other comes close...of course you need to put up with Mono sound...the performances are nothing short of magic!
So, nobody likes the Lindsay's?!
Cpdunn, I find the Linday's unconvincing and out of tune. I hate to say it, but they sound like students to me. Much like the Chillingarians of late- who were good at one time.
Buxter, I agree!! I was waiting for someone to say so! I can't stand them, and Peter Cropper's sniffing is enough to drive me off a cliff.
I'm surprised noone has mentioned the Emerson String Quartet.

Their complete quartets is fantastic. The late quartets are impeccable and exciting.

I would certainly agree that anyone interested in this music should listen to the Busch and also to Quartetto Italiano.
Ah, yes, Dan....too bad some of these "vintage" (makes me feel old!) recordings have been relegated to the dustbin. The Quartetto Italiano was always one of my favorites. Have a lot of their stuff on vinyl, thankfully.