Over time, we've noticed that the kitchen sink faucet cold tap is increasingly getting worse with I'm thinking is water hammer from either limescale buildup or the air pockets in the tap becoming filled with water. When we turn on the cold just a little, it will immediately shut off and make a "bang" noise. To actually get cold water running past a trickle, we have to turn the tap on past half-way. Since this only occurs in the kitchen on the cold tap, I'm guessing the blockage is either within the faucet or on the cold supply branch that leads to the kitchen from the main.

Problem is a loose washer in the cold water side.

Turn off the water at the stop valve under the sink.
Remove the handle, then remove the bonnet nut (the nut around the stem).
Remove the stem and replace the washer on the end.

Harold nailed it. Loose interior part in angle stop. Sometimes it can be fixed if you Close and then Open angle stop. Try it, let us know how it went

Milo

Both experts gave you excellent advises, the only thing I will suggest to shut the water to the house before you remove the stem other wise there will be a mess, good luck.

John