It spins, but doesn't pump
If your washer spins but doesn't pump the water out, the drain line is probably clogged. In many washers, a small sock or other piece of clothing can get between the clothes tub and the outer tub that holds the water. If the clothing gets between the tubs, it may then get into the drain hose that's attached to the pump--or even into the pump itself. If it's in the pump, you need to remove the hoses from the pump and pull the item out.
To remove the sock from the outer tub port, open the washer's main access panel and remove the large-diameter rubber hose that connects the pump to the bottom of the outer tub. Then, using needle-nose pliers, try to grab and remove the clothing through the port.
Sometimes you can't remove the stuck clothing from below. Then you have to remove the agitator, top of the outer drum shield, and inner clothes tub. This isn't easy to do--and you may need special tools--so you might be happier getting a qualified appliance repair technician to do the job.
If the drain line isn't plugged, the problem may be with your pump. Even if the pump appears to be turning, the internal impeller may be broken. If so, you need to replace the pump.
It doesn't spin or pump
If your washer doesn't spin or pump water out but the motor is running, your washer probably has a frozen pump pulley. If so, you need to replace the pump. To check the pulley, remove the pump from the washer and try to rotate the pulley manually. If it doesn't turn freely--if it's frozen or stiff--replace it.
It pumps, but the water returns
If the water that pumps out of the machine goes back into the machine after the spin cycle, your washer may be siphoning the water from a laundry tub that has a slow drain, back into the washer. The usual remedy for this is to improve the draining of the laundry tub. (Is something stuck in the drain?) Also, check for these problems:
If the drain hose reaches more than about 4 inches into the laundry tub, cut off the excess.
If your drain hose is lower than the washer's cabinet, install an air gap/siphon break assembly.
Gertrude, the water must drain for the spin to work. So in your case the problem seems to start with the water not draining. When you move your selector switch by hand, you should hear a click when it gets to the spin cycle. If you can not hear it, it is possible that it is your selector switch that is faulty. Also, check your lid switch and make sure it is working properly. If you have a multimeter, you can check it for continuity. You can also quick check the lid switch, if your washer stops when you open the lid, or continues to work if you open the lid and depress the switch with a screwdriver. the other thing to do is check your check your pump/drain hoses. Look for a kinked hose or some sort of obstruction in the hose. Remember, the missing sock? It may be stuck in there.:-) It is also possible that you have a bad, pump. If you can give us a model number, I can check for a manual for further assistance if needed. I believe the Kenmore washers are actually build by Whirlpool. There are some decent drawings and diagrams on this website. I hope this helps and gets you going in the right direction. Good Luck.