Here is my list of recommended clarinet music:
1) Mozart - "Clarinet Concerto in A, K622" w/Sabine Meyer, live recording w/Claudio Abbado(also has Debussy)
2) Mozart - "Clarinet Quintet K581" w/Sabine Meyer or Richard Stolzman
3) Webber - "Clarinet Concerto" w/Sabine Meyer
4) Krommer, Spohr - "Clarinet Concertos" w/ Sabine Meyer, Julian Bliss
5) "French Works for Clarinet and Piano"(includes Saint-Saens, Poulenc, Devienne, Milhaud) w/Sabine Meyer, Oleg Maysenberg
6) Copland Clarinet Concerto w/Stanley Drucker
7) Corigliano Clarinet Concerto wit either Stanley Drucker or Richard StolzmanĀ
both are good
8) Stravinsky Ebony Concerto w/Richard Stolzman.
9) Finzi Clarinet Concerto w/Emma Johnson. Unfortunately this one isn't performed/recorded often enough thus the selection is limited, but Emma Johnson is pretty decent
10) Poulenc Sextet for Piano, Flute, Oboe, Clarinet, Bassoon and Horn
11) Brahms Clarinet Sonata #1 and #2 w/Harold Wright
12) Brahms Clarinet Quintet w/Harold Wright
13) Brahms Clarinet Trio
14) Schubert - Der Hirt Auf Dem Flesem w/Harold Wright
You basically can not go wrong with Sabine Meyer recordings of anything. She has a traditional, classical, round tone with very little to no vibrato at all. My other favorites are Stanley Drucker, Ricardo Morales, Harold Wright and Richard Stolzman. Stolzman plays with a bit too much vibrato for my tastes, but otherwise really good. Harold Wright's early recordings are very good as well.
FWIW, I would not bother with Thea King on Hyperion label. Her performing is sub par and Hyperion CDs tend to be very pricey. I initially liked it on the first listen when I bought the Brahms sonatas recording but on the second, more critical approach I found that she disappoints. I haven't listened to the cd since then.
Mozart Clarinet Concerto K622 and Quintet K581 are a joy to listen to and if you play clarinet are lots of fun to play as well. Stolzman or Meyer do superb job on these 2 pieces.
Poulenc is a really interesting composer. As a matter of fact, I would recommend everything from Poulenc, not just chamber music. His piano concerto and concerto for two pianos are both outstanding. The French works for Clarinet and Piano cd is great. I couldn't stop listening to it when I bought it.
Anyway, good luck and enjoy the music!