Steering wheel shimmy.....

I had the same problem. Once my new wheels were installed, problem went away. I would have your tires rebalanced and also check to make sure your tire, when mounted, doesn't have a high-spot. One of mine did which the installed discovered. He demounted that tire and spun it on the rim to yield a truer running combination.

what type of wheels are you using?

stock wheels, tires were just rebalanced. 205/60R15

How old are the tires? If pretty old, say over 10 years, they might be coming apart inside.

RRS1,
Check the condition of the top & bottom ball joints as well as the tie rod ends.
When you steer one way or the other the caster loads up the suspension components.
This is because caster creates a jacking effect.
For example if you steer left you are actually jacking up the front left of the car as the wheel is sweeping downward as you turn.
That's why with the more caster, the heavier the steering becomes. Your steering wheel effort is lifting that corner of the car.
If there is any free-play in the ball joints its during the steering movements that you will feel the wheel shimmy as the natural effect of caster is that the wheel wants to be pointing straight ahead.
Your forcing it to turn, but it does not want to be there.
Any back-lash in the steering will then be noticed.
Hope this helps.
regards,
Tony.

That makes sense. I will get it to an alignment shop.

RRS1, adding double the caster via offset a-arm bushings will cover up many instabilities caused by worn tie rods ends, bent links or out-of-balance tires, but I approve of trying the other things before adding more caster.

Deeper into the system, stock steering rack bushings regularly wear out- 5000 miles is about the max life, and it's laughably easy to add zerk fittings to ball joints and tie rod ends to allow regular servicing. Zerks markedly extend their life for pennies. Details on request.

Just had it checked today. Everything was normal as in, set to factory specs, so a little mystery there. It may just be a tire, even though they are MY12 date code.

What's more disconcerting is he said the rack bushing was worn because he could move the tire up and down a little. I just replaced the darn thing! All I know is it doesn't knock when I go over rail road tracks, and there is no play at the boot like there was before.

I did add zerk fittings on the upper ball joints.

I am opening this old thread because I feel my car may be doing the same thing.
I replaced the steering rack washer (with the brass one) YEARS ago.
The tires/wheels do not feel out of balance and the shimmy in the steering is more pronounced when I change lanes. Sometimes going straight at highway speeds (and above) the steering wheel is often rock solid/smooth.