As long as your car is due for inspection, no appointment is actually required. You can simply drive down to any LTA-Authorised Inspection Centre and have the inspection done on the spot. Of course, going during peak hours of the day may require you to wait for your turn, as compared to off-peak hours. Take note also that the opening hours vary for each inspection centre.
You do, however, have the option of making a priority booking with VICOM and JIC inspection centres here (pre-payment necessary), or booking a time slot with STA inspection centre here. These time slots are usually outside of the popular peak hours, and some queuing is still necessary.