DIAGNOSIS You've likely got a burned-out halogen bulb. They're high-heat bulbs and have a finite lifespan, especially if you use the under-cabinet light a lot.
FIX First, confirm it's the bulb and not the switch. Unplug the microwave. The bulb is accessed from underneath. You'll see a clear or frosted lens cover. Give it a quarter-turn counterclockwise to unlock it, then lower it down. Carefully unscrew the halogen bulb (let it cool first if you've used it recently) and replace it with a new 50-watt halogen bulb, type G9. Don't touch the new bulb's glass with your bare fingers—use a cloth or paper towel, as skin oil can cause hot spots and shorten its life.
IF THAT DIDN'T WORK If a new bulb doesn't fix it, the issue could be with the light socket or the control board's relay for the light circuit. That's a more involved repair requiring a technician.
HEADS UP Only use the specified halogen bulb. An LED retrofit might seem tempting, but it can cause electrical issues because the socket is designed for the heat and power draw of a halogen.