Just purchased this car. I was told that it was "running but all of a sudden quit starting." I bought the car; replaced the battery and starter and it still will not start. I just get a single 'click' sound, that is all. All lights work, radio, Windows, ect.. I can actually feel the clicking in the shifter.. Other then that, it has no symptoms. I have cleaned and made sure the signal wire is making good contact.

 

Maybe you should look more into why you can feel the shifter clicking, A vehicle will not start unless it is in Park or Neutral and the position of the shifter may not be properly aligned or sensed.

See if you can get it to start by moving the shifter around into different positions, even see about between gears.
It wont try to start if it doesn't think its in Park or Neutral and the fact that you can feel a click in the shifter is not normal.
The usual no start click or continuous click is a sign of a low battery and the click is the starter selonoid getting enough voltage to attempt activating but not having the power to crank the engine. So I see why you went to the battery and starter first.

However you should have attempted charging the battery it had first and see, a battery regardless of its health will make the starter do the continously fast click sound if it has some power to activate the starter selonoid.
And that click would not be felt anywere unless you touched the starter with your hand while someone tried to crank it over.
As the battery dies the click slows, sometimes a very dead battery can only do 1 click, but your new battery rules that out.

So since you can feel a click its probably that switch clicking, either because the gear lever isn't lining up with the switch or the switch itself is broken.

I think they call it a "Neutral Position Switch"?

The starter was probably fine, and a fully charged battery should make the engine turn over at the very least.
So check out that switch.
(I have heard of the "W" switch for winter mode getting gunky and stuck pressed in, but forget what problems that caused the guy)

 

I ran through all of the gears and nothing changes as I try to start the car. I'm going to locate the neutral safety switch and see if there's a way to test/bypass it.

 

I took apart the center console to find that the noise was coming from some sort of solenoid engaging when I try to start the car. I've had no luck tracking this issue down.

 

TTT. Any assistance would be appreciated

 

There will be a click heard in the center console when you insert the key.


Are you hearing the click when you insert the key or when it is in position III trying to start?

 

It only makes the noise when I try to start the car

 

Nobody else has experienced this?