If it were me, I'd approach this like an "unkown" tail light. Using jumper cables (with either contact tips or alligator clips on the ends), I'd test the bulb - straight off the battery. Then I'd insert the bulb and using a known ground connected directly to the socket and to a bolt or screw, test the light when the switch is on.
Keep moving down-stream on the ground until it stops working and you should find the bad ground. This can also be done with a multi-meter.
I'd done all this on my turn signals and found it to be a bad ground. On my turn signal, I ended up running another wire (for ground) from inside the case all the way down the wiring harness to a good connection under the dash.
