Capacitors and cold or loose solder joints are the most common failure.
Step 1: Caution
Beware of high voltage. Capacitors store energy. They store energy even when the power is removed. Always remove your device from power and discharge capacitors using a resistance probe. Proceed at your own risk!
Step 2: Disassemble
Take your time. Do not rush. Remove the obvious screws and look for hidden screws behind rubber feet, under stickers. Most gadgets are screwed and snapped together. A little wiggle and pressure will unsnap. Separate parts into layers and keep track of the screws.
Step 3: Examing and Repair
In the case below you will see the capacitors are bulging. They are failing capacitors. The 15 volt cap even whistles. After replacing the power worked and this monitor is salvaged with a 1$ part from Radio Shack.
Be sure to replace your capacitors with the same farad rating as the old cap. You can replace with a different voltage as long as the new cap is rated for a equal or higher voltage rating.