The best way to keep any lampshade clean is to regularly dust or vacuum the fabric with a brush attachment to remove dust. Use a microfiber cloth to wipe the wiring clean or any dirt or grime (and dust off your lightbulbs while you’re at it, which use more energy when they are dusty). Different fabrics will react differently to stains. In general, to remove stains on a silk or cotton fabric lampshade, submerge it in a bath of lightly soapy warm water (mild detergent or natural soap works best here), rinse, and air dry. You can try spot cleaning with a clean toothbrush for any stubborn stains, but don’t rub too hard or you might damage the fabric or make the stain more pronounced. A gum eraser often does the trick on a paper lampshade.