With the power off, run through this decision tree from electrician Scott Caron:
(1) Is there black around the socket?
If No, go to (2). If Yes, replace the fixture. It's likely been "overlamped," meaning that 100-watt bulbs, say, were used in a 60-watt-max socket.
(2) Check whether the tab at the bottom of the fixture is pressed down.
If No, go to (3). If Yes, that could be causing a short. Gently lift the tab with a flathead screwdriver back to its original position.
(3) Does the tab look corroded?
If No, go to (4). If Yes, clean with an emery cloth to clear the way for a better electrical connection.
(4) Plug a digital multimeter into a receptacle on the same circuit and see if you're getting more than 120 volts.
If No, go to (5). If Yes, the extra juice is blowing the standard-bulb filaments. Buy a rough-service or 130-volt bulb.
(5) Is there a kid's play space or bedroom overhead?
If No, go to (6). If Yes, excessive vibration could be breaking the filaments. Buy a rough-service bulb.
(6) If all else fails, swap in an LED bulb (making sure to replace any dimmer, too, with an LED-friendly unit). These solid-state illuminators aren't bothered by excessive voltage or vibration, and their low energy consumption won't tax an old socket.