I have a pretty nasty shimmy when going over even small bumps at more than 45mph. The wheel rotates a good inch in each direction. We are in the process of moving into a house so as soon as my tools make it to the garage I'll be investigating this but I just wanted to get some ideas as to what the most likely causes are. It's so bad that I won't take the car on the highway.
My ideas were tie rods or steering rack...possibly sway bar end links. There's a bit of a loose sound when entering drive ways at slow speeds as well.

This is what my wheel does when I hit minor bumps. NON NEARLY THIS SEVERE but to give you an idea.
Jeep Wrangler Death Wobble - YouTube
I don't hear the tires like you do in the Jeep but it is definitely translating to the wheels a bit because the whole car shakes left to right when it happens.
It's a pretty unnerving feeling so any advice is good advice.

 

are you lowered? could be bumpsteer, but you would probably notice it right away, i would check all the steering, tie rods and ball joints and such. and make your your wheels are tight

 

Was previously on Pedders. Now on SS parts. Checked it out today and my lower Knuckle>Strut bolt on the driver side was backing out. Tightened it up to spec and it seems to be gone. I'm not yet convinced that wasn't the effect of something else though.
With that fixed I was able to notice a pull to the left and vibrations from the front while braking. I'll get the rotors resurfaced and swap pads then get an alignment and see where I'm at.
If all goes well with that I may finally be able to put this damn car up for sale.

 

Loctite on strut to knuckle bolts goes a long way