If you have an older whirlpool washer that will neither spin nor agitate and you can hear the motor running (not just humming) more than likely you have a broken coupler. Typically there is a buzzing or semi grinding noise associated with the bad coupler. Here is a video that shows how to replace the coupler.
The coupler is the part in whirlpool direct drive washers that connects or "couples" the transmission and motor together.....there is no belt on these machines. The coupler is designed to fail in an overloaded washer situation in order to avoid damaging more expensive parts like the motor and transmission. This is a relatively easy repair to get your washer back up and running again.