Snapping noise when compressor shuts off

start up capacitors bad on compressor

What's the model number of outdoor unit

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the snap or noise you hear is internal to compressor.

when the compressor is running unplug the ref. and if you hear the noise coming from the compressor then
a ballance spring may be broken inside the compressor.

The compressor will run normally with a broken spring it will just be annoying when it shuts down.

it sounds like the compressor insides are loose and you will be needing a new compressor, but just check and make sure the rubber mounting feet on the outer compressor base where its mounted to the refrigerator base are all tight and the rubber shocks are all good

check te rubber mounts at the bottom of the compressor
the rubber rots after some time and the compressor moves on shutting down as the knocking could be that noise

The compressor is that black casing you see at the bottom of your refrigerator - inside this sealed case is the motor assembly which stits on 3 or 4 springs that act like a cushion - you have a little gap at the bottom where the lubricating oil is present. Your springs have likely worn out and when the compressor shuts off the assembly is banging against the rigid casing make that loud "klonk" or bang noise. It IS normal for compressor to make some faint "tick-tick-tick...tick.....tick" when shutting off, but loud banging, bell sounds, etc, means the assembly is banging on the case from the inside - you can indeed live with the noise, your compressor will function normally, probably for months, or even years - it can also keep wearing out and one day the noise might get very loud (like a very high humming noise) - There is NOTHING that can be fixed, you either pay $400+ to change the compressor or buy a new refrigerator - If you DO replace a compressor usually the replacement will be noisier - Keep a thermostat inside your refrigerator at all time to make sure your compressor is still keeping your refrigerator and freezer cool.

The noise is coming from the compressor and is a result of a broken support spring inside the compressor. When the unit shuts down, the inner casing rattles against the outer casing because the spring is not there to keep them apart. The only fix is to replace the compressor but otherwise it is just bothersome.
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Have someone check the run capacitor of this unit. If the run capacitor is bad over shorted it can cause the unit to run and short cycle on a current overdraw. This makes for an electrical imbalance in the compressor motor. Next step is the compressor is faulty.

are you sure it is the compressor making the noise and what sort of noise is it. if it is the compressor, there is no cure except replacement which is not diy.