I generally tend to prefer live recordings of the music types I listen to most--classical and jazz. Even in the rock genre, the spontaneity of live recordings often outweighs studio wizardry. Besides the added spontaneity of recordings which are not made in snippets and where there is a real live audience to create a feedback loop between performers and audience, live recordings tend to be recorded with less microphones. Less mikes generally means better sound--again, even in the rock area.
I think it is just a fact that good musicians play better when they have an appreciative audience, regardless of the genre of music. And to the extent that even the best musicians will occasionally make mistakes, I'd rather listen to that type of more human recording than the note-perfect productions which most studios feel obligated to turn out.
As far as actual choices for my favorites, I'll first echo a few that have already been mentioned:
Weavers Reunion at Carnegie Hall 1965
Harry Belafonte at Carnegie Hall
Eva Cassidy, Live at Blues Alley
Chuck Mangione, Land of Make Believe
Sinatra 57
Clark Terry, Live at the Village Gate
Someone else mentioned Patricia Barber's Modern Cool. While I also admire that recording, unless there is another version with the same title, that one is a studio recording. As far as live Patricia Barber recordings go, I'd put Companion on the list.
In terms of audiophile recordings, I'd pick two from the Reference Recordings catalog:
Your Friendly Neighborhood Big Band
Red Norvo, The Forward Look
For any Grateful Dead-Heads out there, I'd put Live Dead on the list, as well as any number of the Dick's Picks recordings--too numerous to mention. These days, however, if I want a Dead fix, I usually just turn on the Greatful Dead Channel on Sirius satellite radio.
Also in the rock area, I'll add my "guilty pleasures" recording: Kenny Loggins, Outside from the Redwoods. Fabulous sound (if your system is not balanced too brightly) and rousing performances. I know, some find Loggins (and Messina) insipid, but not me and especially not here.
I'll also mention the Harry James recordings on Sheffield Lab. While not live recordings, they were made direct-to-disc in a single take per LP side and thus have much of the spontaneity of live recordings and have very simple miking, the main mike being a single-point stereo pickup.
As to Jazz in the Pawnshop, my two cents is that while the recorded sound is outstanding, especially for the time, the performances are not captivating. Try the Red Norvo, The Forward Look recording instead. This was Keith Johnson's first recording with his then-brand-new Ampex 3-track tape recorder, and the liner notes say it was made on 12/31/1957. The sound is every bit as fine as on Pawnshop and the musicians--well, let's just say that they play better jazz.