How can I determine the cause of a noisy spin cycle in my INDESIT washing machine?
the motor has 'carbon brushes' which are spring loaded and transfer the mains power into the center of the motor.  When they get worn the motor starts to whine.

These are inexpensive to replace.

The motor in most machines is at the bottom. Making sure the power is off - tip the machine back and keep it safely located in that position.

Crawl beneath to get to the motor which is held in with 3 bolts.  

If the motor has a large canister (capacitor) wired into it then it does not have brushes - the belt is then the culprit.

If it is a 'carbon brush motor' / brush motor then unplug the cable.

http://en.wikipedia.org/wiki/Brush_%28electric%29

Motor out : at each side of the motor the brushes are held in with small springy clips, these can be tricky to remove. If the dark square carbon rod part of the brushes are less than 1.5cm then they are worn and need replacing.