How do you repair the licence plate light on a 1991 Pontiac Grand Am when you have already replaced the bulb?

There are a few other [than the bulb] possible causes for failure of a license plate light to work. That light[s] "rides" on the tail light circuit and its fuse. Since you did not mention a tail light problem, it seems that a failed fuse is not the problem. There are a few other common causes which come to mind: 1. If water, and the road film contaminants which it carries, gets into ANY light bulb socket or connector, the resulting film, or subsequent corrosion CAN create enough resistance in the circuit to restrict the flow of electrical current to the point that a bulb will operate very dimly, or not at all. If this is the cause, a thorough cleaning of the inside wall of the socket and its contacts usually cures the problem. 2. Basically, the same cause as 1, but AT the GROUND point for the bulb socket, which is normally ONE of the two wires coming out of the fixture which is connected to the vehicle frame or sheet metal near the fixture. If corrosion [rust] forms beneath the screw and connector, the result will be the same as in 1. If this is the cause, the correction is to disconnect the ground, clean the corrosion, dirt, etc from the connector, and the grounding area to which it attached, and then reconnect the terminal to ground. 3. Another, not uncommon, cause is that the wire[s] to the fixture have been cut, broken, or pulled apart. Usually, the license plate light[s] are connected to the tail light wiring haresss in the trunk, or beneath the trunk behind the rear bumper. If examination of the wires reveals this defect, the fix is to properly splice the damaged wire[s], or to replace it. j3h.