Why won't my iPhone 4S turn on? And what can I do to make it start up?

Okay so. When I pressed the home button to turn the phone on, nothing happened so I just assumed that it was doing what it does sometimes - randomly get turned off, requiring me to restart it by pressing the Home and Lock buttons simultaneously for about 10 seconds. So I did that. But nothing happened. I tried again. No go. I plugged it into the wall charger then, but still nothing happened. A friend of mine informed me that when he tried to call the phone rang, which is weird because I did not hear it ring, neither did it light up - it simply did nothing. Now it's been an hour and I tried to start it up again using the lock-and-home-button method but that still won't work. Connecting the phone to my computer via USB cable won't work either; the computer won't acknowledge that the device has been plugged in and nothing shows up on iTunes either. So now I've just left it sitting there. I took the SIM out. I don't know what to do.

Anyway, details of my phone - it's about 3 months old, non-jailbroken. It fell yesterday; I remember that, but it was a small fall and it had its case on at the time and it worked smoothly all throughout the rest of yesterday from when it fell until today when it randomly got turned off. Another thing - it does the random turning-off thing quite often, the thing that requires me to restart it using the lock and home buttons. I guess that might be relevant, although like I said, that method is not doing anything right now.

Please tell me what I can do. This is highly inconvenient and irritating.

If it's under warranty, I recommend that you take it back and have it fox or replace. If your friend called you and the phone was ringing,.it means that the phone wasn't shutting off but rather the display was shutting off. I can tell by your post.
This is a mini computer, maybe tomorrow could work fine, but I strongly recommend to have it fix.

It's impossible to diagnose faults with no obvious physical cause on dead mobile phones without using some fairly sophisticated tools.

Your phone is well within the warranty period; I suggest that you take full advantage of that benefit.

If I were you I'd just take it to the apple store or best buy and have them diagnose it for you. I'm not sure if they charge though. Couldn't hurt to ask. Sorry you basically tried everything I would. It sounds like something internal though so it's probably something that actually needs repairing. Maybe when you dropped it, something came lose and it finally broke today. I would say maybe your display just needed to be replaced but that doesn't sound like that's it either because you said the computer wouldn't even recognize it. Sorry if this is no help. But definitely try Apple/Best Buy. Alternatively google search maybe you'll find some forum where someone else had the same problem and had it solved.