Refrigerators and Freezers operate using compressed gas refrigerant which is at a high pressure in copper lines that run through a compressor, condenser, and evaporator. The compressor which is often the most susceptible part of the system may be damaged if you have frequent power brown outs or momentary power outages of just a few seconds. o see if the compressor is damaged pull the refrigerator out from the wall, confirm the fan in the back of the unit is running when plugged in, there should be a cover (often of cardboard or plastic with simple screws that covers the fan and compressor. The compressor is will be a device that will be shaped sort of like a black egg like shape. with tubing going into and out of it. When the refrigerator cycles on the fan should start and if you listen closely to the black egg it will make a deep rumble (or growl sound) If the compressor does not make any sound after a few minutes 5 to 10 then it is most likely burnt out. If it runs and the fan is operating. Are the coils clean? or are they clogged with dirt or pet hair? In either case unplug the unit. If the coils are clogged carefully using a non scratching brush and without bending the fins clean out the dust, dirt, hair. a vacuum crevice tool is good to use. If the fan runs, the compressor runs, and the coils use to remove the heat from the interior are already clean then the unit most likely has a leak in refrigerant and will need to be charged to attempt to locate it depending on the age of the unit or if it has been moved recently (could have been damaged in the move) refrigerant leaks are akin to looking for a needle in a haystack especially if the unit is older as a leak can occur inside or outside the unit or in the walls of it.
How compressors get damaged. When first plugged in attempts to start immediately, but if the power was just turned off and back on then pressures exist that don’t allow it to start, but the motor still tries and fails then tries again, until they equalize to the point where the compressor can start again. This action will cause the compressor to burn out and require replacement. Same with ac units, which is if you have a power outage I always tell people to turn off the breaker to these items until the power is back on and stable or you may be replacing these devices well before their time.
Leaking may refer to refrigerant gasses may be leaking and if so someone licensed in refrigeration would be needed to check the system, use gauges to verify pressures, and add the refrigerant gas. The refrigerant type is not able to obtained in the quantity needed if you aren’t licensed in refrigerant systems. Also the gauges and other tools needed are quite expensive to purchase but are necessary to repair the unit.