Hello there! I`m a clarinettist of 7 years, and I know TONS about the clarinet (thankfully). The two upper trill keys can be used in many different ways for instance: let`s say you want a better in-tune throat Bb...push the A key (don`t push the register key) and hit the bottom of the two upper trill keys. Your Bb will be more in tune (on most clarinets) and work better for playing in large sections or ensembles; for quicker runs, use the standard Bb, though. Also, for the notes B and C finger the standard Bb (thumb key and A Key) and hit the lower of the upper trill keys for B and hit both the upper trill keys for C. Also, when fingering the high C (Register and thumb keys only) to trill from C to C#, D, or Eb trill with the bottom trill key, the bottom of the upper trill keys, or the upper of the bottom trill keys. ALSO also, to hit the highest C (C4) you finger: R T 1--|1--Eb and to get the C# simply add both the upper trill keys, then finger the high D (D4) by fingering the same as E3 or A4.
The trill keys can also be used to finger the throat F#: hit the thumb key and press the lower two trill keys: better in tune F#...also makes for better chromatic scales.
Hope I could be of some service to ya!
Ryan T.
7-year clarinettist/songwriter
Senior - Twentynine Palms HS
Drum Major - TPHS Wildcat Marching Band
Concert Master - TPHS Symphonic Band
Clarinettist - Pasadena City College Tournament of Roses Honor Band