Actually, the turn signal is the problem... What year is your car? GM turn signals on our cars are a prone problem area. The turn signals are slathered with what appears to be bulb grease from the factory. The smoke you are seeing is the grease finally getting hot and smoking.
Its a pretty easy fix though (well depending on what year car you have) If you don't have airbags its pretty straight forward. All you'd really need is a steering wheel puller, needlenose pliers, a torx bit for the screws that secure the turn signal switch, and your favorite set of sockets.
Once you have the steering wheel off and the switch is visible, just take out the torx screws and the entire switch will pull off of the column. separate the two halves of the switch (they pull apart - don't loose the ball bearing) You will see two sets of contacts with three copper strips each. wipe the grease off with a rag or maybe some TF tuner cleaner (local electronics store) and clean your contacts with a pencil eraser (just like the old slot cars). bend your contacts slightly inward for good contact and coat them lightly with some dielectric tune-up grease, and reassemble.
you'll be good to go.
