The answer to 'sink won't flush' is that a clogged drain is the most common cause. This can be due to debris, hair, or other buildup blocking the drain pipe. Symptoms may include slow draining, gurgling sounds, or water backing up in the sink.
To troubleshoot, you can first try using a plunger to dislodge the clog. If that doesn't work, you may need to use a drain snake or auger to clear the blockage. If the clog is deeper in the pipes, you may need to disassemble the drain to access it. In some cases, the issue could be a more serious problem like a broken or damaged pipe, which would require calling a professional plumber.