From dirt's standpoint, motorcycles tend to be a large collection of cozy little hidey holes all flying in loose formation, that are perfect gathering spots for dirt, grunge and grease. If you're like me, and you'd rather your motorcycle didn't turn into a mung condo (mung being the technical term for the type of dirt that likes to collect in hidey holes) then you'll need to spend some time scrubbing and the easiest way to that is with a scrub brush of some sort.
For big, gross jobs I prefer to use a parts cleaning bristle brush, available at any auto parts store or a cut down paint brush. Parts cleaning brushes are generally round and stiff, and come in a number of sizes which makes them perfect for cleaning things like sprockets, or digging through the goo on the underside of the bike or for cleaning between the spokes and around brake calipers.
Paintbrushes are too flexible to really worry at caked dirt, but cut them down by a third or more and they'll develop enough back bone to be of some use. Wrap a few layers of duct or masking tape around the metal binder to prevent scratching anything and it'll be good to go. I buy the el-cheapo ones at the big box store and toss them after a few uses. In standard lengths paintbrushes work really well on things like engine cases, radiators, oil coolers and wheel rims and are handy for reaching in between the cylinder fins. You can also use a clean one, or a stiff hobby or artists brush to remove waxy residue from the spots you can't reach with your polishing rag, for example from around the edges of a tank badge.
Small jobs need smaller brushes. Old toothbrushes or purpose-made industrial nylon brushes can be used to scrub carburetor bodies, petcocks (boy, does' that sound dated) or any other small semi-delicate parts, and of course they're great at removing dried up wax from all those spots you missed with the paint brush. An acid brush, you can find these anywhere plumbing supplies are sold, will in some instances work even better than a tooth brush. Cut the bristles down by half and you've got a nice small semi-rigid scrub brush that'll chase the mung out of anywhere you can reach with it. Cut it down a bit further and you can use it to apply anything from gasket sealer to Semichrome. Because acid brushes have metal handles use a few winds of tape to prevent any uh-oh moments whenever you're working around something you'd rather not scratch.
Lastly there are some situations that call for the use of a small, relatively soft wire brush so I always keep a few small brass or stainless steel brushes handy to remove corrosion or polish small parts. The brass ones work great when used with metal polish to shine up things like axle nuts or other crusty hardware, while the steel ones are terrific for removing rust and heavy corrosion. Most hardware or auto parts stores keep them in stock and you can usually buy a fistful for a couple of bucks.