This post is closely related to the post on using the register key, the post on bass clarinet underblown multiphonics and also to the post on Eb clarinet underblown multiphonics. It might be worth reading those, as well as the posts on spectral multiphonics for Bb, bass and contrabass clarinets, and Bb and Bass dyads to have a complete overview of this aspect of clarinet playing. (Basically read all the posts about multiphonics? There are quite a few more…)
If you’re looking for a nice example of a piece that uses these (plus a rather wide range of dyad multiphonics), check out Martin Rane Bauck’s Kopenhagener Stille. The section where he makes exclusive use of these multiphonics starts at 8:18 (but the whole piece is terrific!).
As usual these are in written (transposed) pitches. The fingering for each multiphonic in the first two sets is actually the same as the fingering for the top note, so you can notate these by just writing u.b. or underblow. If you’re worried about confusing the clarinetist, include a link to this page in your legend.
As with the Eb and bass clarinet equivalents of this technique, these are best performed at quiet dynamics.
The last few here are a bit theoretical most of the time, but you can hear the low note, just (you might need headphones.
Going upwards from the C sharp, the multiphonics are quite fragile, and a bit louder, but very effective:
There is also quite a wide range of alternative fingerings for these, should a) something not really work very well or b) you be looking for a slightly different colour. These may well be some of my favourites, actually…