Just last week I had a "no power" situation on a brand new LG electric dryer. The problem turned out to be a disconnected hot wire inside the dryer.
To troubleshoot your refrigerator, you'll need a voltmeter. Start at the cord going into the refrigerator and check for 120 Volts. Unplug refrig and perform continuity checks on individual wires. It's a tedious process but it's what we technicians do all the time. It's what I had to do to figure out the dead LG dryer.