All of my past experiences have been related to fan /bearing shrieks.
Give us the brand and model and the area that the noise seems to be originating from, and is it also having a separate freezer section ?
If it has a fan (some do) do the blades rotate without hitting anything? Is the fan working ok (i.e. bearings)?
The compressor is physically isolated from the metalwork by rubber bushings. One may have failed. Similarly, a lot of the freezer pipework is left 'floating' in the air - is any part of it touching the chassis/frame? Simple massaging (slight bending) of the pipe work will shift it out of the way.
If the noise is from the compressor bearings then you need a new compressor.