Common problems
DNS server errors were very common in earlier PlayStation 3 firmwares, but since the introduction of firmware 2.20, these errors have been few and far between. When, or if, you do receive this error, there is one simple solution, allowing you to use the PS3's online capabilities. When configuring your PS3's network settings, input everything manually instead of allowing the system to do it automatically.
Another annoying problem that might happen is if your controller becomes out of sync with your PS3. This means your controller does not connect to your PS3. Simply plug your PS3 Bluetooth controller into the PS3 USB port and push the middle PlayStation button on the controller itself. Your controller will resync itself.
Infrequent Problems
Sometimes your PS3 will have no display when you turn it on. This is caused by a number of things: if you switch TVs, switch from HDMI and component cables, or if cables are loose. Check the cables first to make sure they are properly connected. If that is not the problem, turn off your PS3 and turn it back on, but when you turn it on, hold the power button until you hear three beeps. This will reset the PS3's display settings and allow you to configure.
Serious Problems
When your PS3 will not power on, check all the wires, making sure they are not cut or burned. Make sure they are all connected properly into the TV and into the PS3 itself. If you get through all these things and there is still no power to your PS3, plug a different device into the outlet to make sure it isn't the outlet itself. If the outlet is not the problem, get in touch with Sony Computer Entertainment of America customer service. More than likely you have something wrong inside the PS3, and a replacement could be in order.