I think for many doublers the clarinet’s left-hand “sliver” key seems useless or problematic. For example, the sliver key is easy to press by mistake when intending to cover the middle and/or ring finger holes. And even when reaching for the sliver on purpose, it’s easy to accidentally cover part of the ring finger hole, producing an E-flat or B-flat that is flat and stuffy.
The left-hand sliver also lacks any real analogue on any of the other common woodwinds, so its use is a technique that doesn’t transfer easily from another instrument. Flutes, saxophones, and standard bassoons don’t have any key in that spot. The oboe has a trill key there, but its usage isn’t similar. Among the standard band/orchestral woodwinds, only the contrabassoon has a key positioned here that is used in a similar way to the clarinet family. Especially for saxophonists, the right-side fingering is much more familiar.
The Woodwind Fingering Guide (still the best fingering source on the web) lists three E-flat/B-flat fingerings in its standard clarinet fingering chart, with only the right-side-key fingering marked as “basic.” The left-hand-sliver fingering is described as a “Chromatic and trill fingering,” to “use in combination with D4 [D below the staff] and A5 [A above the staff].” (The “one and one” fingering using both index fingers is also listed, though it might perhaps be better relegated to the “alternate” fingering chart.)
Occasionally I’ve run across the attitude that the sliver key could perhaps be removed or wedged shut to prevent accidental venting. I think this would be a waste, and all clarinetists of an intermediate level or higher should get used to using this key as an equal partner with the right hand key—not merely as an alternative for rare occasions.
Here are a couple of examples from well-known solo repertoire where the left-hand sliver makes sense:
In this example from Debussy, the E-flat occurs four times, each time preceded by E-natural, and followed in three cases by D—this is the aforementioned “chromatic” usage. Among the three most-common E-flat fingerings, the sliver is clearly the best technical choice here. While one-plus-one works equally well (from a technical standpoint) moving from E to E-flat, E-flat to D is compromised by having to move two fingers, each on a different hand, each moving in a different direction; the intonation issues of one-plus-one further complicate its use. The right-hand-side fingering is less hairy, but does involve moving fingers on both hands simultaneously (a maneuver which should, of course, be possible, but which carries unnecessary risk of mistiming).
This excerpt from the first of Finzi’s Five Bagatelles highlights some other use cases. In the second measure of rehearsal number 4, a well-trained clarinetist will use the two lowest right trill keys for G-flat, which would make a transition to one-plus-one or right-side E-flat unnecessarily awkward, particularly since the E-flat then moves right back to G-flat. The sliver key allows for crisp, tidy finger movements. In the last two measures of this excerpt, the sliver key again constrains the fingers to similar motions in adjacent fingers of a single hand.
If you are new to the use of the left-hand sliver (or just a little rusty), I suggest conditioning its use for both E-flat and B-flat with slow, deliberate scale practice. Start with the chromatic scale and B-flat major, then look for appropriate places to use it in other major and minor scale material.