My poor kitchen sink faucet has been neglected lately. And neglected means that some hard water stain crusting started to pop up, especially in the joint where it's difficult to clean.
However, just like my trick for cleaning hard water stains off my shower door tracks, I just soaked a paper towel in distilled white vinegar, wrapped it around the worst area and let it sit for about 30 minutes. Then, I used an old toothbrush to get in those hard to reach spots and get the gunk out.
You can use the vinegar-soaked paper towel to clean the rest of the faucet that doesn't need to soak since vinegar is perfect for getting rid of water spots. This method also works great for getting that hard water crust from the bottom of the faucet where it meets the counter.
If you have granite counters, be careful about getting vinegar on the counter, since it's a mild acid and can damage the seal.