Happens during agitate as well as spin.
Happens even w/o any chlothes in it.
When rotating it looks like it is rolling over a small bump.
I jat turned it by hand to feel for "rough spots." When I did so, I looked between the gasked and the basket and noticed a pile ao plastic shavings. They have come from the plastic shroud that rings (and guides?) the metal basket.
I am very handy and now quite broke with 4 kids.
Can I fix this myself?

Found a PDF of the repair manual on line.
Took the machine apart.
The SS basket is supported entirely from the rear bt a cast (stainless?) steel. It has three stout arms that radiate from the center where the bearing is
Two of them a cracked all the way through. One of them looks like it had been that way since Day 1 because it was enlarged and caked with grime like the rest of the exposed surfaces.
So, I thnk it was a defective unit when acquired 10 years ago.
Needs new basket @ $450.
Needs new front half of tub @ $70.
New machine costs $1000

Sorry about the typos... Is there a way to edit a question?
In any case, I called Sears - no help.
Called Whirlpool and kept asking for a "supervisor."
They offered discounts on the parts so I am out only $300 including shipping.
Good enough.
Hope I will remember how to out it back to gether in "7 to 10 days."