I have the same problem except mine is the master shower on second floor. The master bath works fine as well as the sinks. I think it is a frozen pipe under the floor. Need help on options....

We have only had it happen once during a really cold spell in central NC - 8 degrees for about 5 days straight which is very unusual here. The water pipes to our master bathroom sinks run up an outside wall to the second floor and they froze. Nothing else was affected. We just waited it out until it got warmer.

I assume you don't have cold or hot water running to your sink. You may want to go into the crawl space under your home at the kitchen sink area and make sure the pipe has not burst under the house before it turns to go up into the wall. You can call a plumber to have them look at it. You could also put heaters under the house - just don't leave the house with them running.