The upper portion of the agitator (Kenmore Series 80 top load)stopped spinning as if the thread has stripped. What parts need to be replaced?

387780 does not seem to be a valid model number.
If you can locate the model number enter it here at sears Parts Direct.
Parts for Brands Craftsman GE Kenmore Whirlpool and more

Link to videos about washer not spinning.
top loader kenmore washer won spin YouTube

well i would recommend repacing agitator and agitator doggs ... get complete model number off washing machine data place got to sear.com and select partsdirect enter model and order parts they will send directly to you...

check the dogs in the agitator take off the cap it should rachett if it turns free replace the dogs in the agitator

There is a video at www.appliancehelponline.com/servicemanual.html that will walk you thru the replacement of the agitator dogs, this is a common problem with this style washer.

If the bottom agitates and not the top, the problem is likely the 4 "agitator dogs" designed to allow the top agitator to turn only one way.

This assumes that you have a basic Kenmore / Whirlpool top load, direct drive, neutral drain (it doesn't spin until the pump removes most of the water) machine.

First, see the Sears parts site for your washer:
http://www.searspartsdirect.com/partsdirect/index.action

Enter your model number (###.######) and you'll see a list of major sub-components with diagrams and parts.

Look at the section on the Sears site for Agitator, basket and tub.

See the part "Dog, agitator "

Remove the fabric softener unit, lift the top agitator cap (use a screwdriver) and you should see a plastic part in the center of the top agitator with a 1/2" square drive.

Remove the drive and lift off the top agitator. Turning it over you should see the 4 "dogs", replace these and the top agitator should turn in one direction only.

Replace the 1/2" drive, the cap and fabric softener unit and the top agitator should work.

Also see the following illustrated description of the above.

http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=136&osCsid=0da65cea0e338fe930b1fde153a243f0

If the top portion of the agitator (called the "auger") is no longer spinning like it should, you may have worn agitator dogs. The following link explains how to access and replace them:

http://www.fixya.com/support/r823955-replacing_agitator_dogs

This is a very simple an inexpensive repair. The part number is included in the link I provided. If this does not correct your problem, or if you have any questions, please let me know. I hope this helps you.

If the upper portion of the agitator is not working, this is commonly caused by worn agitator dogs. The following link explains:

http://www.fixya.com/support/r823955-replacing_agitator_dogs

Yes...there is such a thing. These small plastic pieces have been the source of many trouble calls and only cost a few dollars to replace if you know how to do it. Replacements can be ordered on line at searspartsdirect.com, appliancepartspros.com, or repairclinic.com. Use your model number for the search criteria and look under the "Agitator Parts" section. The link also give instructions on how to remove the agitator. This is a very simple repair, that only costs a few dollars. If you wish not to order on
line, use the part number provided by the Sears website and go to your nearest appliance parts retailer. This is a common part they will have in their inventory. If you require further assistance, please let me know. I hope this helps you.

If it will not spin either, you most likely have a broken drive coupler. This machine is "direct drive" and has no belt. As techs, we see this several times during our work week. Otherwise, if you have the "two piece" agitator, you may simply need new "dogs" in the agitator. if the machine spins fine, fill it only on the smallest load size with no clothes in it. When the agitate cycle starts, take note if the bottom section of the agitator is going back and forth. If it is, and the upper part does not "ratchet" then you just need to rebuild the agitator and that is an easy job.

your agitator dogs are stripped there about $10.00,remove cap to replace. easy,need 1/2 to 3/8 adapter or 7/16 socket depending on model. and your coupler is broken about $20.00 its what runs yous washer its like your belt under neath between motor and transmission , less cloths max water will stop it from happening again

If it agitates and drains. leaving your clothes clean but soaking wet. It is most likely the transmission/gearbox. the motor is working otherwise it would not agitate. it is stuck in agitate mode and cannot "shift" into spin mode. new transmission will cost 140-170 and takes 2 or so hours to replace depending on skill level. total repair cost parts+labor $300ish you may want to ante up for a new machine at this point. if you cant to the labor yourself.
part# should be this: tho double check using your own model#
Part # AP3096209