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There are several things to check for, fill valve, ...pump, ...center seals and tub bearing bearing, ...fill air break and surrounding hoses, ...suds saver, are all common leak area's. For center seal and bearing replacement, see that page. With the use of plastic drain hoses on many machines today, this has become a common leak trouble maker. The drain hose can rub the wall or a pipe and this will rub a hole in the drain hose.
Check belts, if they look like this, replace them. Check lid switch and lid switch plunger assembly. Newest Maytag washers have a different style of a lid switch, it looks like this. This "new" style switch is bit confusing to test, this here might help you out. Sometimes things get caught between the tanks as well. You can test a motor with a cheater cord...to test a 2-speed motor and to test a 1- speed motor. The motor pulley (#17) can also break apart and the wash may not go. The motor pulley can be changed from the bottom of the washer.
Make sure nothing is caught on top of the inner basket or between the inner and outer tank. The motor sits on a roller and track assembly with springs to help keep the belt tension. This track gets rusty and dirty and may need cleaning, the track/glides also need a lubricant to keep the track moving freely. Sometimes the center bearing can get worn and bind on the transmission, this can, and will stop the washer from spinning and may also burn the belt. Try to turn the belt by hand, if it is super stiff...possible the center bearing has seized.
One thing to check is the spin bearing, the bearing is inside the top of the brake package. There are damper pads glued on the base that the washer assembly sits on, these pads are a "buffer" between the suspension assembly and the base. If these pads wear out, usually a grinding type of noise is noticed, especially in the spin mode. How to change my damper pads.
Check for a seized transmission, if you cannot turn the belt by hand the transmission may be seized. There is also a drive lug that may need to be adjusted, the drive lug is under the center (Trans) pulley.
It is a good idea to always check the drain hose hose for kinks, the pump also can get objects caught in it ( socks, lint ball, etc ), lastly check the pump it self. Sometimes the impeller inside the pump can break free from the shaft and the pump will turn but wont pump the water. Remove the hoses and put your finger inside the pump so you can touch the impeller, with your other hand turn the pump assembly pulley, if the pump impellor has broken free you can hold the impeller from turning.