A clean vase makes a bigger difference than you'd think. Here's the most effective method, plus the mistakes most people make.
Pour out any old water and remove any dead flowers or debris.
Pour 1/2 cup of vinegar and 1/4 cup of rice (or salt) into the vase. The combination scrubs the inside as you swish.
Cover the top and shake the vase vigorously for 1-2 minutes. The rice scrubs the inside without scratching.
Empty the vase, then wash with warm soapy water and a bottle brush. Rinse thoroughly.
Place the vase upside down on a drying rack to air dry completely before storing.