Unclogging a kithcen sink that is backed up is similar to unclogging any other drain. The variable may be whether you have a two sided sink or not. The tools needed will be a plunger, a rag, and maybe an auger or snake (for stubborn clogs).
Unclogging a Single Basin Sink
Make sure there is at least 4 inches of standing water in the sink to start. Use a plunger and plung forcefully 10 to 12 times. Remove the plunger to see if the sink drains. If not try running the garbage disposal (if you have one) and then repeat the plunging process. If this doesn’t work you will need to auger the drain.
Put a bucket underneath the sink drain trap to catch the water. Disconnect the trap slip nuts and remove the sink drain trap. Manually push the auger cable 6 inches or so into the drain entering the wall. Slowly begin snaking the drain until you push past the clog. You should feel a little resistance up until you get past the clog and then should be able to feel when you clear the blockage. you can pull out the clog by running the snake counterclockwise or just flush out the blockage. Next, put the trap back on the drain and test out the sink. If the sink does not drain freely, repeat the drain augering until you have successfully cleared the blockage.
Unclogging a Double-Sided Sink
Fill both sinks with at least 4 inches of standing water. Use a rag or a closed strainer to close off the garbage disposal side of the drain. Use your plunger to plung the other side of the sink 10 to 12 times. If this doesn’t clear the blockage switch sides and plung the other side 10 to 12 times. If this doesn’t unclog your kitchen sink then complete the augering steps list above. The procedure is the same in a double-sided sink as it is with a single basin sink.