Washer stays in cycle and doesnt advance to additional cycle

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

This is almost always caused by a bad time. The solution is to replace the timer. What happens is the timer has a small motor on it that turns the timer, if the motor burns out or the timer gears wear the timer doesn't advance like it should.

Can you provide the exact Make, Model and Kenmore model number? ..........

Kenmore model numbers use the format ###.######## with the first three digits of 110 for Whirlpool manufactured machines.........

The model number on top load washers is usually found by lifting the lid and looking above the tub.

The Fixya numbers are Stock Numbers. Series numbers (70, 80, 90) only designate the number of "Bells and Whistles" that a washer has and don't reflect the model number.

See the following for where to find the model number:

http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=21&osCsid=c73289e58485c8752675ae1a35ff796b

Since basic top load direct drive, top load electronic and front load washers are very different, the actual Model Numbers are necessary to correctly diagnose the problem.

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.

The first thing I'd do is see if the washer will continue it's cycle in a different mode. i.e. Try to start it in a different cycle, or try to move the timer knob to the spin cycle.

Following is a Whirlpool / Kenmore installation and maintenance PDF on how top load, direct drive washers work.

https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/f7e2147fa98b28cf85257122007475cb/$FILE/L-55.pdf

It has a troubleshooting section as well as detailed descriptions of how this type of washer works.

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.

This appears to be a timer / timer motor problem.

I'd alto look at the Whirlpool / Kenmore direct drive installation and maintenance manual for testing the timer.

https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/f7e2147fa98b28cf85257122007475cb/$FILE/L-55.pdf

Assuming that this is a basic top load, direct drive, neutral drain washer it seems as if the problem is likely in the timer.

The transmission and draining are controlled by the motor which gets it's power through the timer. If the timer isn't advancing once it gets to the spin cycle area, the motor keeps getting power, the machine keeps spinning and only by pushing in the control knob will the motor stop.

Here are a couple of places to look at how the machine works and parts for a basic Kenmore washer.

Sears parts site:

http://www.searspartsdirect.com/partsdirect/index.action

Enter your model number and you'll see a number of sub catagories for the washer components.

Whirlpool Installation and maintenance PDF.

This is a public domain Whirlpool manual describing how the washer works and a good troubleshooting section.

https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/f7e2147fa98b28cf85257122007475cb/$FILE/L-55.pdf

If the washer will fill and drain, but won't agitate or spin, you may have a broken motor coupling. The noise of the motor engaging, lack of agitation and ability to still drain are the usual symptoms. This is a common fail item with this model washer through normal wear and tear. Newer versions of the coupler are more resilient and resist breaking. The following link explains how to replace one:

http://www.fixya.com/support/r385250-replacing_motor_coupling_kenmore_top

The motor coupling is located between the drive motor and the transmission and takes the place of a drive belt. They do wear out and can break. The replacement part number is included in the link I provided and is an improved upgraded coupling that is more resilient. It should last longer. If you have questions, or require additional assistance, please let me know. I hope this helps you.

A replacement coupler can be purchased on line at appliancepartspros.com for about $11. Just use the part number in the link and "Search by Part Number" for your search criteria.

Now...a washer that fills and doesn't do anything, or stops right after the wash cycle and then does not advance to the rinse cycle and drain is commonly caused by a lid switch malfunction. The lid switch is usually mounted under the washer top casing, on the right hand side on most top load models. It is activated by a plastic piece on the lid (called a "Lid Strike") through a small hole on the rim of the wash tub opening. In many cases the lid strike breaks off the lid, or the lid switch becomes loose and no longer makes contact. Inspect the lid to make sure the lid strike is still intact, and then inspect the mounting screws (adjacent to the small opening on the wash tub rim) and make sure they are snug. If you determine the switch to be defective, the following link explains how to replace one:

http://www.fixya.com/support/r385198-replacing_lid_switch

If the switch requires replacement, a new one can be purchased on line at searspartsdirect.com, pcappliancerepair.com, repairclinic.com, appliancepartspros.com. Just use your model number as your search criteria. All these sites offer competitive pricing, so shop all them for the best price.

Take note of your symptoms:

1. Washer fills, may or may not agitate, will not advance to the rinse cycle, and will NOT drain - lid switch.

2. Washer fills, does NOT agitate or spin, but WILL drain - motor coupling.

If you still have questions, please let me know. If the suggestions I have listed do not match your washer symptoms, please post back and elaborate more on what the washer is/is not doing. That way I can provide you with better assistance. I hope this information is helpful to you.

I purchased a new timer, installed and solved the issue.

BAD LID SWITCH-MIKE

If pump works fine or still pumps out water, then it is a timer malfunction. Replace the timer and you should be fine. To locate a part number for your timer, refer to www3.sears.com with the model number off your washer.