I bought this 1997 Chevy Cavalier secondhand with 86k miles on it and have taken great care of it. At 100k miles after a heavy rain I noticed the windshield wipers stopped in the "up" position. Basically the wipers were always 180 degrees out of phase. When they were supposed to be down, they were up and up when they were supposed to be down. So whenever the wipers were turned off, they stopped right in my field of vision. If I put the delay on low, they would stall in my field of vision as well.

I traced the problem back to the wiper motor. I bought a rebuilt wiper motor for around $75 from a local parts shop and it was a little tricky putting it in, but it was do-able within a few hours. As I remember, the general steps were:

1) Take off wiper arm bolt caps, bolts, and then lift wiper arms off (the construction of the wiper arms made this pretty difficult--they don't pull right off. After attempting to pull them straight off, I tried to use a battery terminal puller, but this did not work either. I found the best way to remove them was to "wiggle" the arm back and forth as I eased it off the mounting bolt). 2) Remove bezel surrounding wipers. This was fairly straightforward. There were a series of tabs holding the bezel on under the hood which must be pulled out and possibly a few bolts too although I don't remember. 3) Remove bolts holding in wiper motor and unplug and remove wiper motor. There were three bolts which must be removed to get the wiper motor off the frame and wiper linkage. 4) Open wiper motor and check contacts for corrosion. This is really optional. I thought it may have been terminal corrosion, but I didn't find any.
5) Plug new motor into the car and "zero" it. To make sure the new motor is at the end of a phase, plug it in, turn on the car, turn the wipers switch to "on" and then "off". This will ensure that the motor is maximally displaced to the wipers being down. 6) Without turning motor, attach to linkage. As I remember it, there were 3 tabs which must be lined up on the linkage before the motor is put back on. You may have to take out the linkage completely to accomplish this. If so, be careful of a plastic mount which is easy to break which seats the linkage. 7) To save yourself the headache, reattach the wiper arms before replacing the bezel and turn the wipers on. Make sure everything is lined up ok and that they are working. 8) Remove the wiper arms and reattach the bezel. This can be a little tricky and you want to make sure the rubber molding of the windshield isn't damaged here. The best way in many cases is to push the bezel down (it's made of plastic) as you push it forward under the molding. 9) Reattach wiper arms.

All in all, an annoying problem, but one which can be fixed by people who are handy enough. I'm by no means a car expert, but by doing it myself I was able to save a few hundred dollars.