Looks like the fixture may have gotten hot when the bulb burned brighter as its filament experienced an internal short just before it burned out entirely. Have you taken the fixture down from the ceiling to see if the wires leading to it are fried/show any signs of overheating? Does the switch on the light fixture feel like it has good detents, or is it feeling loose/worn/damaged?
Using a voltmeter, do you see 12 volts present at the fixture wiring when the fixture SHOULD be working? If so replace the fixture (highly recommend going to LED fixtures ...) If no voltage present, then you have to look further back into the wiring run supplying the lamp.
A package of 5 lamp fixtures to replace the ones you have only costs about $42 as seen here: