Product review (5 stars): Washer now spins.  This part fixed my problem completely
2/14/14 UPDATE -- It's still workin' like new!

APPLIANCE:
Kenmore 500 top loading washer

PROBLEM:
Washer suddenly would no longer spin during rinse and spin cycles.  Therefore, clothes would be soaking wet at the end of a wash, and would not be ready for the dryer.

TROUBLESHOOTING:
We believed it was not a motor failure, since I could hear the motor trying to turn the tub.  Googled the symptoms and found out clutch failure was a common issue with the Kenmore 500 top loaders.  Ordered this replacement part on Amazon (~$8.95 plus expedited shipping; total still was under $18).  Received it in 2-3 days.

INSTALLATION:
I should point out there are many helpful videos on youtube.com now for appliance repair.  Watch more than one since they are not guaranteed to be accurate.  I found at least two that showed how to take apart a washer similar to ours.  They also showed how to put a new clutch in.

TOOLS USED:
channel-lock pliers to remove supply lines
pail to catch water from the supply lines that you are going to disconnect
several dry towels
flat tipped screw driver to remove top control panel and clips
3/8" drive ratchet
6" ratchet extension
7/16" socket to remove agitator bolt
1/2" socket to remove motor bolts
pliers (normal slip-joint type) to remove hose clamps if desired
needle-nose pliers to remove clips
nut driver set (optional but very convenient)

RESULTS:
Our washer now works like new.

COMMENTARY:
I could see how worn our old one was when I took it out.  These clutches are somewhat under-built for the job; basically the design is like a drum brake, but the pads are not very substantial. The new one is similar to the old one, so it may wear out again someday.  However, this part fixed our problem immediately, so if it does break again I know I could probably swap in another one in about an hour.