A sinking fund is a way to save up for expected expenses. Basically you are setting aside the money now, because you know that you will need it later. If you are planning carefully and you are aware of your usual expenses you should be able to avoid most financial emergencies (saving your emergency fund for the possibility of job loss) and save in advance for what many people consider emergencies. You should prepare for these unexpected events, beyond just having an emergency fund in place.
You can have a sinking fund for car repairs and another for home repairs or remodels. You can also have a sinking fund for vacations. You can also have a sinking fund for medical expenses. You may have a sinking fund for the next car you are going to purchase. You can also plan one for any major purchase, so that you can pay for it with cash. Setting up your sinking funds now will make it easier to reach your other savings goals. It can also reduce your financial stress.