Open up the cabinet doors, and start blowing some heat towards the back wall. A hair dryer would work too.
How to thaw frozen pipes
All the facuets (and baths) in our house work fine for both hot and cold EXCEPT the one in the kitchen sink, which has no water flow at all, and the utility sink in the laundry room, which only has cold water. It is about 10 degrees outside and we live in Maryland.
But would that account for laundry the utility sink in the laundry to only have cold water? The kitchen and utility sinks are not located along the same wall.
Since it has been so cold in the Mid-Atlantic region for the past few days, I would bet on a frozen pipe. Are the supply lines for the kitchen sink on an exterior wall?
I have been fighting this problem for years. I am still working on it. Two things you can do to delay the freeze. Insulate the pipes, insulate the walls and seal the air leak. The cold air blast from a air leak from outside cold air on the pipe is like a "anti" blow torch. Seal any cracks. Wrap the pipes in insulatings, as thicj as you can find the insulation. Insulate the back side of the sink cabinet wall if that wall is the outside wall. Insulate the walls of the cabinet. If opening the cabinet doors is undesirable, then put a low wattage lamp under the sink. The heat will warm the space.
All that is assuming the freeze location is at the pipes under the sink. For my case, the freeze is at the pipes that ran under the floor before they reach the sink cabinet. I know because I can clear the clog pipe with a hair dryer at the space at the basement foundation floor and the joist cavity where the pipes ran (my kitchen is built on an enclosed porch).
If your pipes are ran inside an outside wall, they are at risk. There is not much space bwtween the pipe and the very outer layer of the wall for any substancial insulation. Rerouting the pipe may be your only option.
The freeze at the kitchen and the laundry may or may not be the same freeze. It coud be two separate freeze. Are they from the same feed? Maybe the freeze is further back at the source. Any pipes running along/near the foundation wall? There are lots of air leaks alng the sill plate (is that what it's called? where the wood rest on the concrete blocks?). A constant cold blast will freeze the pipe.
Your pipes can freeze in the middle of your home. I have seen this many times. When I open the wall to fixed the burst pipe, there is usually a huge cold draft in that wall. Key is to stop the draft in your plumbing walls.
Many of our customers are using the Redytemp to prevent their pipes from freezing. Alaska and Canada included. As long as the water in the pipes is moving it can't freeze. If the outside temp is rediculously low then they just set the Redytemp to a higher temperature to compensate for the abnormal condition. Redytemp's temperature range now includes lower temperatures for the specific purpose of pipe freeze protection. Most would rather spend the little extra to keep warm / ambient temperature water circulating throughout their pipes than to risk the high cost associated with the obvious damages which can occur when a pipe burst due to freezing. You might want to take a look at it.