Modern Method

Mild, scratch-free brass polish
Plain dishwashing soap
Make sure to use a scratch free solution and sponge or cloth

For Heavy Staining

Mix one cup vinegar with 4-5 tablespoons of salt and let blend soak on the surface of the singing bowl for 3-5 minutes. 
Rinse thoroughly with water so no salt or acid remains on the bowl.
Rub the bowl down with a soft cloth to dry.
Apply a thin coat of beeswax or olive oil.
Make sure the bowl is dry before use.

Traditional Method

In Nepal, singing bowls are traditionally cleaned with rice straw ash. The ashes are mixed with mustard oil, and the singing bowl is rubbed with this blend until the surface is clean. The mustard oil leaves a thin protective layer on the bowl similar to a polish or wax.

Things To Consider

The color of the signing bowl will change over time, depending on how often it is cleaned. If you wish to maintain the golden shine, you should clean it regularly, about every two months.
If not cleaned, a green or brown film called Patina will develop on the surface of the singing bowls. This is produced by oxidation over a long period. This does not affect the sound of the singing bowl. The aesthetic quality of the bowls over time is a matter of personal preference. 
Cleaning does not affect the sound of the singing bowl.