Beyond the most basic requirements, most clarinet players carry a number of optional but extremely helpful accessories. This beyond-the-basics tool-bag includes items of convenience, maintenance and best-practice. While they are all portable, a couple of these things will not fit inside a standard case. Generally players either carry an extra satchel for supplies or have a case cover (weather cover) with extra storage space.
Clarinetists are often plagued by moisture condensing inside various tone holes. I don't know of any way to prevent this. However, a single piece of cigarette paper makes for an effective remedy.
Cleaning out an afflicted tone hole has four steps. First, swab the clarinet. Second, blow through the tone hole to remove the bulk of the moisture that has settled there. Third, swab the instrument again. Finally, use a piece of cigarette paper to tamp away the remaining surface moisture. Drying the area around the hole seems to help slow down the speed at which clogging recurs (moisture apparently attracts additional moisture.
Cigarette paper is the preferred material to use because it doesn't disintegrate like most papers. And unlike most cloth, it doesn't leave any lint or other unwanted particles that might gunk up the tone hole. Note that most brands of cigarette papers have a small strip of glue along one edge of the paper. Be sure to tear off or otherwise avoid bringing the glued portion in contact with the instrument, especially pads.
Water does a much better job of moistening a reed than saliva. Most professionals use a small cup of (preferably) luke warm water to soak their reeds for a minute or two, rather than soaking the reed in their mouth.
There are many small, lidded containers that suffice. Plastic film canisters are a popular choice, although, one visitor to this site has commented that these containers might have harmful chemical residues. Tupperware makes a small, reasonably water-tight container that works well. The main advantage of the Tupperware container is that a stand clip is made specifically to fit it. The small, metal cup holder clips onto a standard Manhasset-style music stand and holds the water cup off the floor. Many catalogs and music stores often consider this an oboe supply, so it may not be listed in the section with clarinet accessories.
Music Stand Shelf
While it's unusual to carry a stand shelf around, it is a very convenient item to use at home with a Manhasset-style music stand. A stand shelf clips onto the shelf (lower edge) of the music stand, and provides extra space to put pencils, mouthpiece cap, reed holder, swab, and various other items.
Clarinet pegs (or other type of instrument stand) are almost a requirement for orchestral players who must often make quick changes between B-flat and A clarinets. Pegs come in a wide variety of styles. In general, the heavier, larger ones make for the safest stands, They are also the least convenient to carry around.
In recent years, light-weight plastic pegs have become available. These offer convenience (light-weight, easily portable) but trade away stability. My favorite among the portable pegs is made by K & M. It's three legs fold up for compact storage. When extended they provide reasonable stability, even on carpeting. The smallest peg I have tried is the Pak-a-Stand. It folds up small enough to store inside the clarinet's bell in any standard case. Pak-a-Stand works OK on hard, flat surfaces. However its short legs do not give it any stability if the instrument is bumped. Pak-a-Stand does not do well on carpeting.
The final item in the Beyond-the-Basics kit is a small, jeweler's style screw driver. Tiny screws and springs used throughout the clarinet have a habit of coming loose or undone at the most inconvenient times. A small flat-head screw driver is all that's needed to avert what could otherwise be a disastrous playing situation.
The accessories discussed here, while not absolutely necessary, are standard tools for professional players. Having them with you often makes playing easier, and, in some cases, can save you from a tough situation. Any accessory that can make that claim is well worth having.