If the pipes are frozen leave the cabinet doors open in order to allow heat from inside the house to warm them. If frozen, you are all ready doing what you can. If the attic got too cold, you could have overhead supply pipes in the attic frozen that supply the kitchen. You may want to open an attic door and allow some heat to rise into the attic. If the temperature rises and you still do not have water at the kitchen sink and everything else is working then you may have something else going on.
If you know you do not have frozen pipes and your cold water is running everywhere else. If the pipe is frozen, this will not do anything. If you have a single lever faucet there is a cartridge in the faucet that controls the flow and mix of hot and cold water. If you are getting hot water but not cold water then the cartridge could be clogged or faulty. The best thing to do is remove the cartridge and clear the cold water input to the valve body. What you can do is turn off the hot and cold shut off valves under the sink. Remove the cartridge after removing the lever handle and retaining nut or retaining pin. You may need to pull straight up with a pair of pliers to get it out. After removing, with somebody helping and a towel over the open valve body, you can turn the shut off valve for the cold water back on briefly and see if you are getting water flow. If so, good. That means the cartridge is defective or clogged and needs to be cleared or replaced.
Now, if you do not get cold water flow from turning on the cold water shut off valve briefly, then you need to remove the cold water supply hose from under the sink valve and insert it into a bucket and then turn the shut off valve on and see if you get water flow. If you get flow, good. The clog should be somewhere in the cold water supply side of the valve body and you need to get it cleared out, Compressed air is good if you cannot pick out the clog. Now if the supply hose does not give you cold water flow into the bucket then you need to remove the supply hose from the cold water shut off valve and verify it is not clogged. If the hose is clear then you need to move on to the shut off valve. If you can get a large cup or bucket up next to the shut off valve you can open it up and see if you get water flow. If you do not, you will need to shut off the cold water supply to the house. You will need to remove the shut off valve and replace it with a new one. With the valve off, check and verify that the pipe nipple or extension coming out of the wall is not corroded or scaled closed. If it is, common with galvanized pipe, you will need to have the pipe nipple or extension removed and replaced with a new piece. Install the new shut off valve and make sure it is closed. Now, turn on the main water supply and allow the cold water pipes to refill throughout the house. Allow other fixtures to run until you have a strong stream of cold water without spitting. Now, you can run water from the new shut off into a bucket or large cup to flush the valve. You can have the supply hose installed to do this easier. Reconnect the supply hose under the sink and put the cartridge back in. Use plumbers silicone lube on the O rings on the cartridge and you can lightly apply the lube into the valve body. Re-install the holding nut and lever and you should be back in business.