How to fix the static screen on external monitor?
I have seen the static on the Apple Cinema LED Display connected to a Mac Mini Late '09 using mini-DVI to Mini DisplayPort Kanex converter.

What helped me was the order in which I connected the devices after the problems occurred. Not sure what monitor and what converter you use and whether these steps would help; however, what I learned is that these converters are very (VERY!) sensitive devices.

SHUTTING DOWN


Turn Macbook Air off (NOT just sleep, better reset PRAM and NVRAM, make it forget the terrors about the converter)
Disconnect the converter from the Mac
Disconnect the monitor from the converter and wait like 10 seconds
Turn your monitor off (pull from the power outlet, let it forget about the converter completely as well)


RECONNECTING


Connect the monitor back into the outlet
Connect the converter (my converter has the
USB female and male cable, which are extremely important to connect as
well, otherwise it acts weird)
If you DO have the USB, plug it in into the Mac first
Plugin the Mini DisplayPort into the Mac
Turn your Mac on


The reason for all this is to make your ports read and initialize the devices in the right order.

Also, keep in mind how to disconnect the devices correctly.

Disconnect the converter from the Mac first (Mini DiplsayPort)
Then wait a bit and pull USB if you have it.

Do not disconnect the monitor from a converter leaving the converter in your Mac. For some reason it causes problems when I connect the monitor back. My Mac won't see that the monitor is plugged in. Think of it as a converter having the drivers and its own soft running on your mac, so when you disconnect the monitor from the converter, Mac might not fully recognize that the monitor isn't there since the converter won't pass the signal saying, "Hey, the monitor is fully disconnected, stop all services." It will keep running the processes until you disconnect the converter from the Mac. So when you connect the monitor back, it might not react.

I do not like converters. I always try to avoid them as much as I can. So with the new Macbook and USB-C thing, I don't know... Hope they will build them better and smarter.

Good luck!