Clarinet
Microtonal fingerings for normal clarinet
Quartertone cross fingerings are widespread (and apparently standard technique in French orchestral playing today), but certain pitches right above the break have no fingerings.
Richard Eldon Barber has made available this fingering chart for a 31-tone JI scale.
AFMM's Pitch publication has fingerings up to 72 per octave.
Also worth mentioning is the scheme for making a small xenharmonic clarinet by connecting the mouthpiece directly to the lower joint. This fits only on certain clarinets, and it produces a macrotonal scale with no high register.
Actual microtonal clarinets
All clarinets are tuned to an inexact 12EDO. It might be more accurate to describe them as 19ED3 (19 equal divisions of 3/1), since they overblow at the twelfth.
Quarter-tone clarinets
A quartertone clarinet was designed in 1937 by Fritz Schüller. It uses two parallel tubes, one slightly longer than the other, with a single key to switch between them.
Bohlen-Pierce clarinets
The Bohlen-Pierce clarinet project by Stephen Fox (instigated by Georg Hajdu) has so far resulted in the building of a soprano clarinet in the BP scale (13ED3); the scale seems ideal for clarinet in particular.
Several compositions have been written for Bohlen-Pierce clarinets: by Canadian composers Owen Bloomfield and Todd Harrop; as well as by German composers Georg Hajdu, Peter Michael Hamel, Sascha Lino Lemke, Fredrik Schwenk and Manfred Stahnke. Links to the former composers' audio streams can be found on http://www.transpectra.org/audio.html; and links to the latter composers' audio streams can be found on http://mmm.hfmt-hamburg.de/index.php?id=konzert-programm_1362008.