Hi ! First you can pull up the stopper and clean out the drain from the top end , it can get really nasty, lots of hair , hair products , soaps, all kinds of stuff . If that doesn't help , if you have an old hot water heater tank , sometimes it can be coming from that , you will notice the smell after running the hot water .
Depending on the design of sink, the overflow can hold a small amount of water at the bottom where it T's in to the main drain, which can become stagnant. Run water down the drain and you'll force some air up the overflow (to make way for the water coming down), which will have that stagnant smell.
To diagnose this, plug the sink and begin filling it; you shouldn't get any musty smell at first because there's no air movement. Once the water level hits the overflow drain, you will start smelling the musty smell for a while because the water is displacing the gas, which wants to rise above the water and so will move up into the bathroom.
If this is the problem, you can ameliorate it with some foaming pipe snake; pour it down the overflow drain and it will clean out any caked-on gunk which contributes to the smell, and which may be trapping the water. The real fix is to make sure there's no "damming" effect of construction defects at the bottom of the overflow drain (a lip of porcelain, issues where the overflow meets the metal drain downpipe, etc).s there can be nasty things in there that float so they don't go through the trap, and this forces them out the drain pipe.
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Here is a video using a Zep product that might help.
Dump baking soda in it then white vinegar flush with boiling water be careful it'll get really fizzy do a little at a time when you first get started to make sure it is not clogged up & slow draining. This mix will open it up. If this doesn't work just use commercial drain cleaner gel or even clorox gel toilet bowl cleaner. after it's cleaned out put really hot vinegar in it every few weeks to keep it maintained