Kenmore washer will not spin
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:
Enter your model number and you'll see a list of major sub-components with diagrams and parts.
The Kenmore / Whirlpool top load uses a reversing motor to initiate the pump for draining and the transmission to spin.
This is the Whirlpool PDF on installation and maintenance of direct drive washers. See the section on the transmission.
WHIRLPOOL INST AND MAINTENANCE INCLUDING HOW TRANSMISSION WORKS
If everything else works OK,, including the drain, and the timer moves, the problem could be in the transmission / gearcase.
If you lift the lid, put it down again and the washer begins to spin, the problem is likely in the transmission "Neutral Drain Assembly".
Those cams/gears etc. in that assembly keep the washer from spinning until all of the water is pumped out and then go into spin mode.
Apparently, by opening and closing the lid, torque from the motor starting throws the cams into the right gears so that the washer will spin.
Ours had the same problem, no spin unless the lid was raised and lowered or we shut off and re-started the cycle.
After looking at the transmission parts list (available on a number of sites) and on the Whirlpool site (Whirlpool makes Kenmore) we noticed that the one of the cams in the neutral assembly which is supposed to have a small spring, didn't on ours when we tore the transmission apart again.
The Sears site for your washer may have a sub-component section on the gearcase with a part number for the Neutral Drain Assembly repair kit.
Found the part number for a kit (Whirlpool part number 388253 Neutral Assembly) for $15.00 and replaced the cams and gears.
In order to get the transmission out, you have to remove the cabinet, the fabric dispenser and both agitators. The top agitator uses a plastic 1/2" drive bolt. You don't have to remove the drum.
You can also see this site for removing the transmission.
ILLUSTRATED DESCRIPTION OF REMOVING CABINET THRU TRANSMISSION.
Lay the machine down (after disconnecting power and hoses, remove the pump (two clips), the motor (two screws hold two clips on the motor), lift and set the motor aside, remove two 1/2 inch bolts for the motor mounting plate and three 1/2 inch bolts for the transmission. Pull the transmission, remove the clutch (there is a spring clip that holds the clutch on the main agitator shaft, remove the 8 screws carefully the transmission holds 14 - 15Oz of oil, remove the clip holding the main spin gear on and check the plastic cams.
If the cam that should have a spring doesn't, or the others look damaged, or the main spin gear is damaged, the washer won't go into spin mode.
If that's the problem, install a Neutral drain kit, put everything back together carefully and the problem may be solved.
With ours, all told, it cost less than $30 (including new 80-90 weight gear oil) vs. about $170 for a new transmission.
The job is a bit complicated and messy but not too difficult.