I dropped my iPhone 4S and it works fine?

I dropped my iPhone 4S from about a height of 12 inches.
I didn't pick it up straight away, but I noticed that the app I was on was still running fine.
Yet the right hand side of my screen was 2/3 grey. Up in the top right hand corner it looks like bleeding (black). While on the 1/3 of the screen left it's green(ish) but the phone is still completely usable to text/ring/use Wi-Fi/3G.
I turned the iPhone 4S off and now when I turn it on, the 1/3 of the screen I could see is completely filled with green. Meaning I can't tell if the iPhone is actually turning on or not.

Any advice?
(e, g) Take it to a shop for a new screen? How badly is it damaged, etc…

Get the screen fixed. The description you gave seems to imply that the rest of the components apart from the screen are fine. If you want to know whether the phone has booted up or not. First press the power and home button together for about 15 seconds. Now power on the phone how you would do it usually. Press the silent rocker downwards. If you receive the feedback (vibration) then the phone has booted up.

Wao… It is working it means its your good luck. So in this case you should be happy. But you should be attentive for future. It is safe just because of less distance. So be careful in future.

Get the screen change. If you are not able to seen the screen properly. As you said above it seems every thing in phone seems to be working properly the hardware and apps. Only the screen is damage. It is better option to change the screen.