Try checking the fuses. If ok, then it is a wiring or electrical prob. You might also want to check your ignition switch. it may not be making contact properly. I found this happening on my1991 cavalier also. If I would twist the ignition like I was going to start it again, but not far enuff to grind the starter, everything would start woring again. this would sometimes last a long time and other time not so long. So, with that said I would place my money into the ignition switch.
Dryness will cause the wipers to vibrate or jitter against the glass. If there is not enough moisture on the windshield (for example, a light sprinkle of rain) the wipers may move with a jerky motion. Also, the wipers could be very old, and the wiping rubber/vinyl could have lost some of its softness and flexibility. If you mean vibration when they are not active, the wind against the windshield could cause them to flutter if they are not properly seated. A clean windshield and new wiper blades will usually cure the vibration.
From the way that question sounds, there is nothing wrong. When you spray washer fluid on the windshield, the wipers are supposed to automatically come on for probably 3 or 4 swipes, then turn back off - it's a convienience thing when you need to clean your windshield. If your wipers are coming on automatically at random times, then you probably have a loose connection at the switch.
Your wipershaft assembly needs replacement. What is happening is it is drawing much more power then it normally should to run your windshield wipers which is causing excessive heat in the wires running to the motor which can burn up dust, wire casing and other stuff which causes the smoke. This could result in a fire so it would be wise to replace it soon.
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