Novus 1,2,3 works great!
I decided to try a plastic polish on my monitor. I know it doesn't sound right, but I am not the first to try this. There is a thread over on arcadeotaku where a couple of guys tried removing scratches from monitors. Some used T-cut and others used Novus with almost identical results. Since T-cut does not seem to be available ion the US, and I needed some plastic polish to remove some scratches from my cab, I decided to try Novus.
Before - you can see the reflections of the scratches around the lights on the monitor. Every inch of the monitor looked this way actually - you just can't see it in the photo without the light.
After treatment
I decided to use it on the plastic bezel as well. Great results, both the monitor and the bezel look new again.
If you want to try it, Novus 1,2,3 can be purchased on egay - 8oz bottles for $28 shipped. They have 2oz bottles as well, but I have several cabs I wanted to use it on so I went with the bigger ones. Also pick up some of the micro fiber cloth packs.
Its kinda funny because you use it in the opposite order as
3,2,1 or
1,3,2,1 Novus 1 is a cleaner, Novus 2 is a fine scratch remover, and Novus 3 is a heavy scratch remover.
If you want to see how to go about using it there is a decent tutorial on youtube where a guy uses it to remove the haze off his headlights. However, in the thread I found on arcadeotaku, one of the guys used one of these. It really makes much shorter work of it.
As you can see, I just stuck a rag in where you would normally load sandpaper. Occasionally I would wipe the screen down and check for areas that needed additional work. The monitor took approximately 20-30 minutes. The Bezel took about half that long, maybe less.
I am sure you could use Polish people to remove scratches from your monitor but I am going with some sort of rotary tool like aznxtc suggested. 
