We should all know by now that once you complete 15 Taxi Fares in a row, you unlock "Taxi Shield" which means all Cabbies found in Liberty City driving around are completely BP (Bullet Proof).
However, there is a glitch that was brought to my attention about an AP (All Proof - BP/DP/EP/FP ; Bullet, Damage, Explosion, and Fire Proof) Cabbie being *temporarily* obtainable at anytime during the game. I say temporarily obtainable because once you put this beast in the garage, it reverts back to only BP. The garages in CTW eat every special property a vehicle possesses except its BP property. So this AP Cabbie is only good for having fun with before putting it into a garage. You can use it on missions, you can use it for police pursuits, you can use it to ram cars and just do fun things driving around, etc. Just don't put it into the garage. But even if you do, who cares, because you can always get another very easily at any time.
How to Obtain It:
Getting it requires absolutely no luck or skill. In other words, it's very, very easy to get. Anybody can do it, literally, and it can be done at any stage of the game. All you have to do is have completed the Taxi Side Mission with a Gold Medal. Now, just hail for a Cab to come pick you up and take you to a destination on the map. Once you are driven to your destination and get out of the back seat of the Cabbie, just quickly run forward and jack the Cabbie from the guy who just drove you to your destination. This Cabbie is AP and completely indestructible. The only ways it will be destroyed is if you tip it upside down or ditch it into the water where it will become un-retrievable.
Enjoy your new AP Cabbie!
EDIT: Confirmed it does NOT work on Nintendo DS. Now we just need to see if it works on PSP, but I doubt it.