If your pipes are plastic, a manual plunger (the classic suction cup one) remains your safest bet, but how you use it really matters a lot: fill the sink with lukewarm water until the plunger is completely submerged, then press without lifting the plunger from the bottom and make firm but controlled vertical movements. It's not about brute force - around 15-20 firm pumps are already plenty, then you pull it off and see if water starts flowing. If it doesn't work the first time, repeat a couple times with patience.
Here's the thing nobody ever mentions: before anything else, put a wet cloth over the overflow hole (that little hole on the sink wall) if there is one. Without that, the pressure disperses and you'll struggle to break the clog. Then, when you pump, focus on the main basin - don't be an idiot and pump like crazy because you could actually loosen a fitting under the cabinet, especially if your pipes are already a few years old.
If after 3-4 attempts you don't see any improvement, stop there. At that point the blockage is deep (limescale, hair, who knows what) and a pressure plunger like one of those boot-pump ones risks breaking everything. Better to pay a plumber than fix pipe damage - it'll cost you more than an emergency call, trust me.