I never understood how the PS3 controller is recharged, I could never get it right without using PS3. Here are my variations of the failures I've observed:

1) Recharging on a PC that has been turned off, but with USB ports are still powered won't work. The controller LEDs show nothing
2) Recharging on a PC that is turned on, the controller won't charge until OS initialises the controller
3) Sometimes when doing 2), the controller still won't charge until I press the PS button
4) Sometimes when doing 3), the controller LEDs simply flash rapidly trying to sync with PS3 via bluetooth
5) Sometimes when doing 4), unplug the USB cable, the LEDs still flash rapidly, plug in the cable again still doesn't change this
6) Sometimes when doing 4),  unplugging the USB cable stops the LEDs from flashing rapidly, but when plugged back in, LEDs don't flash at all
7) Repeating 1)~6) in any combination sometimes make it recharge again
8) When 1)~7) fail, I've just learned that there's a reset button on the back of the controller, it makes the controller charge again

I've tried with all sorts of power sources, PC/laptop/TV, and quite a few USB cables. Why is recharging the PS3 controller so difficult? No other USB devices I own can match the complexity of getting this controller to recharge.

I've googled a lot to find answers, I've seen other people asking about the same problem, but the most common answer I've seen were "get a new controller", "replace the battery, it's dead". I think both answers are unacceptable, either the controller is defective, or PS3 controllers are simply crap and buggy.

Another problem I had was pairing the controller with PS3 via bluetooth. When I first got my PS3, I plugged the controller to PS3 via USB as instructed by the manual. The controller was quickly recognised. I then unplugged the USB cable to see if the controller can now via bluetooth. Nope, it kept on flashing rapidly. Plug the USB cable back in, play around in the settings, assigne the controller as player 1, unplug it again. Same crap. Out of ideas what to do, I rebooted/shutdown PS3 a few times and tried a few other things, suddenly it started working. Till now, I still don't understand how I got it working.

I asked other friends who also own PS3s, they told me pairing the controller is as simple as plugging in vis USB, let PS3 recognise it, unplug it and it should just work. Not so in my case, but at least once it's paired, I never have to do it again.