2003 Ford Taurus vibrates when I brake but not all the time. Sometimes it shudders very violently when coming to a stop. The steering wheel and it feels like the entire car does this. There is no squeeling or sounds that indicate brakes are going out. Please help

If it tries pull the steering wheel back and forth, it is brake rotors. If it just vibrates threw the body and steering, it is the right side wheel bearing.

If you are not a heavy footed braker... then you have an ABS module problem. There are several sites on line that will do an exchange for a rebuilt one for around $100. If you are a heavy footed braker, your car is behaving normally, Just don't press so aggressively and the vibration won't happen. ABS brakes are supposed to shudder and vibrate when activated, they apply pressure to your brake calipers at roughly 15,000 times a sec, off and on to prevent the brakes from locking up and the car loosing control. In a panic stop, press a car with ABS brakes ALL THE WAY TO THE FLOOR as hard as you can, use the steering wheel for leverage and push the pedal down a bit more, RIGHT BEFORE IT COMES TO A COMPLETE STOP.

Only feels vibration when braking and there is no fault in the engine, the problem is in the brake rotors , Feels vibration in the steering wheel, when applying the brake ?

usually vibration while braking means he rotors are warped and u need to replace or cut [rotors are cheap enough that replacement is usually recommended -but also with front wheel drive vehicles a bad hub[wheel bearing]or cv axle can cause a vibration / noise from front

I would start by checking the fuses in the fuse panel. On the 2003 Taurus it is below and to the left of the steering wheel near the brake pedal.

Trouble code P0420 means: Catalyst System Efficiency Below Threshold

This is unrelated to the noise when turning sharply or the grinding when braking/brake light.

Most of the time, this vibration is caused by warped brake rotors.
It's also possible that the steering or suspension joints are worn.
These should be checked when the brakes are inspected.

Might just be a loose connection on the power steering hoses. You'll need to get under (BE CAREFUL) your car or get it up on a car hoist. You or a mechanic, should be able to trace where the leak is and either tighten the connections or replace the hose. Cold weather brings on leaks ! !
I don't think the brake vibration has anything to do with it. Brake vibration usually means the brake rotors are warping. Common problem. Have you have your brakes checked lately? Might be time for new pads and rotors.
Good Luck

check rotors

sounds like spindle/axle may be bent!