Oldest Recordings that sound “audiophile”

Jo Stafford/Art Van Damme Quintet-Once Over Lightly.
Columbia CL 968 Mono 6 Eye 1957.
Found a mint copy(jacket and vinyl) at a thrift store for 49 cents.

Is not comparable to most audiophile recordings, Recorded in mono it does have that beaming effect but the sound is very natural.

Miles Davis - Kinda Blue 
Dave Brubeck -  Jazz at Oberlin 1953 , the Finney Chapel as well as Brubeck makes that album. 

Some of the very best stereo recordings were done in the late 1950's.  It is a bit sad that the state of the art has not advanced that much.  For example, a number of 6-eye Columbia's from that time are incredible sounding, such as Ellington's "Blues in Orbit" (1959); Brubeck's "Time Out" and Goodman's "Benny in Brussels (1958).  Another beautifully recorded stereo recording from that era is Armstrong's "Louis Armstrong Plays King Oliver."  I use these as demonstration records.

Some standard audio demonstration recordings from the 1960's include "Belafonte at Carnegie Hall" and the Weaver's "Reunion at Carnegie Hall 1963."

There are plenty of mono records that sound great, but, being mono, they give away their age.  Still, who is not amazed by Rollins' "Saxophone Colossus" ?

For classical, I like the Cozart/Fine recordings for Mercury. 

I agree that the Nancy Wilson, Frank Sinatra and Nat King Cole recordings on Capitol sound really good.  I had an opportunity to hear a master tape for a Nat King Cole recording and it was unbelievably vibrant, and realistic sounding.

There is something to what Teo_audio said about having tubes all the way through the recording, mastering, cutting and playback chain.  I have a few Acoustic Sounds reissues that claim that tube gear was used throughout, and although I have no way to attribute the sound quality to this particular aspect of the reissue, I do note that these reissues sound terrific.