Part number of starter compressor

SOURCE: Kenmore (253.51399104)side by side making buzzing/humming noise

It will rattle slightly, this is normal. The ones that are burned out will sound like they have gravel in them, you can defininely tell something is wrong.
The problem with diagnosing Kenmore fridges with this problem is that there are different manufacturers. Yours is built by Frigidaire and probably has a Tecumseh compressor. We find more failed compressors with the Tecumsehs than we do with the models that have Embraco compressors. The models with Embraco's generally have more relay failures.

You should not hear the buzz/hum at startup. You will need to replace the compressor, which will be replaced at no charge to you if it is less than 5 yrs old.
If it is older than 5 yrs, you can have it replaced, but it will be almost as expensive as a new fridge. You can buy an aftermarket start device called a 3-in-1. This has a start capacitor built in to give it a little more boost on start up. I don't recommend using them unless it is a short term fix until the compressor can be replaced or if it is out of warranty and there is nothing to lose anyway.
Thanks/Mike

SOURCE: Refrigerator buzzes & clicks

The only thing a capacitor does is 'kick' the compressor to start it. There is a box on the compressor containing the start relay. If it is defective or dirty, it might buzz. With the power off, find it, remove its cover and use some canned air to blow off any dirt inside. If that doesn't fix it, replace it. If there is a fan down there, be careful...and be sure it is working.

SOURCE: can't find compressor

The compressor is LG part number 2521C-A7256 which they have in stock. I believe this unit has a 7 year warranty on the parts of the sealed system (which includes the compressor) but this is most definitely not a do it yourself repair. Only someone certified to handle refrigerants, has the proper refrigerant recovery equipment, and charging equipment (and knows how to use them) should ever attempt to replace a compressor. Compressor failures are rare, and the unit should be diagnosed by a professional service technician, as most often your problem will not be the compressor, but some other issue. Please do not rate this solution as anything but a FixYa lowers my score.

SOURCE: Fridge won't start, tries to but clicks off after a few seconds

compressor is broken if you are under warranty you can get it replaced. if not you should buy another one cause compressors are expensive to replace. costs about the price of a new refrigerator. you can try getting a used one from classifieds or craigslist.com

SOURCE: Help w/ Refrigerator Compressor Overheating?

I had a similar problem yesterday (Maytag side by side unit 5 years old, however this apply to most brands). All operation seems normal other then that the refrigerator was not cooling, the air that was blowing inside the refrigerator was room temperature and not cold air. The compressor was also very hot to touch (could not touch it for more then 1-2 seconds) and didn't buzz or hum as in standard operation. I also heard few clicks from the compressor every time I plugged / unplugged the AC to the fridge. Replaced the overload/ relay device with a new one I got at a local appliance parts distributor ($30) and MAGIC. The unit went back to work as soon as I plugged the AC power. The relay/overload unit is on the side of the compressor and that?s where the electric wires are attached to. The unit turns the compressor on and off. The unit is attached to the compressor side with an "electric outlet" like three holes and three prongs. It might be held tight so you can pry gently (between the unit and the compressor body) with a screwdriver. Once pulled out, release the wires connectors that are attached to the relay/ overload unit. You might need to use pliers as they are tight fit. If unit smells like burnt or if you shake and hear rattle then it?s bad. Find a local appliance parts distributors and call to see if they have part for your model/ type. Remember to unplug AC before starting to work on the fridge and good luck.