Figure A--Standard lampshades (those that are flat with a papery feel) are very easy to clean with a dry sponge, which can be found at paint and hardware stores. This sponge feels like a dry pencil eraser and works like one, too. Wipe the shade from top to bottom to remove lint, dog hair, dust, etc. No water is involved. Pleated shades are a little tougher to clean, since the pleats can catch dirt and lint in the folds, making the dirt hard to reach. Use a toothbrush to get inside the folds.
Figure B--Fabric lampshades, especially those with fringes, can be a challenge. To clean stains or spots in the fabric, use spot remover and a soft cloth to rub out the stain. Add soapy water if the stain is stubborn. These shades can be vacuumed, but adjust the "bleeder" so that the vacuum doesn't pull too strongly on the shade. The bleeder is found on the wand of the attachment--just open it up to reduce the vacuum pressure.
Figure C--An immersion method can be employed to clean lampshades. This can be done in the bathtub or in very large containers. Fill one container with soapy water and another in rinse water, then simply immerse the shade in the soapy bucket and rinse in the rinse bucket. Allow to air dry on cloth.