Shop-vac suction at a sink drain: where does it work?

I’d lean toward a wet-rated vacuum as a first try only if you can make a reasonably tight seal without forcing anything or risking damage. If the pop-up, overflow, or second basin makes that setup awkward, cleaning an accessible trap is more direct - you can inspect it rather than hoping suction reaches the obstruction. If the trap is clear or the clog seems farther along, neither approach may resolve it.

If only one sink is slow, a blockage near that sink or in its trap seems more plausible; if several fixtures are backing up or draining poorly, I’d suspect a shared line farther along. Neither clue proves the location, but I’d be more inclined to inspect the trap in the first case and avoid relying on sink-level suction in the second.

A pop-up stopper usually isn’t the main obstacle if the drain opening is accessible; the bigger issue is keeping air from escaping around the vacuum nozzle. If an overflow or the other basin is open, suction may leak there too, so gently covering those openings can help. I’d use a soft, non-marring material and stop if it takes force to make a seal - if the setup stays awkward, opening an accessible trap is the more straightforward option.

I’d avoid both methods if several fixtures are backing up, since that can point to a blockage beyond the sink, or if the trap is stuck, damaged, or hard to reach safely. A single cautious attempt is different from repeating suction or dismantling without improvement; if the trap is clear and the blockage is farther down the line, repeating either approach is unlikely to help. At that point, getting help seems more sensible than risking damage or making a mess.