Locate the shutoff valves for the pipes. Sometimes there is a valve in sight, while other times the only recourse is to turn off the water to the whole house. Leave the valves on unless you find a leak.
Open the faucets in the sink slowly to release any built-up pressure. Leave them open until the pipes are clear of ice. As the water flows by any ice buildup, the heat in the moving water will help to melt the ice.
Open the cabinet doors under the sink and turn the heat on in the room. The warm air will gradually warm the pipes and melt the ice. To speed up the thawing process, use a hair dryer to blow warm air directly on the pipes. Be sure to have a bucket ready in case there are any leaks.
Slowly turn off the faucets and check along the pipes for any obvious leaks or drips. Turn the shutoff valve clockwise to shut off the water flow.