The answer to 'why is my sink won't flush' is that there is likely a blockage in the sink drain. This could be caused by a buildup of hair, food particles, or other debris that has accumulated over time and is preventing the water from draining properly. Common symptoms include slow draining, gurgling sounds, or water backing up in the sink.
Some common solutions to try include using a plunger to dislodge the clog, using a sink snake or auger to remove the blockage, or using a chemical drain cleaner to dissolve the buildup. However, if these DIY methods don't work, it's best to call a professional plumber who can inspect the pipes and use specialized tools to fully clear the drain.