Steering wheel "rumbles" and shake

A bit more information please. Could you please describe this noise in more detail? Is it a metal on metal grinding noise or more of a humming, rumbling noise? is it a single 'Thump' as you brake or a steady noise that changes with speed? Does the vehicle pull to one side as you brake? Does it happen when you turn tight corners?
The steering shaking means there is a vibration somewhere between the road wheels and the steering wheel. A common cause for the noises and vibration is a warped brake disc. Even the smallest warp can cause a lot of trouble.
Try this. Pull the Park brake ON, put wedges under the back wheels and the gearbox in Neutral. SAFELY jack up the front left side. You only need to get it high enough so the wheel is a couple of centimetres off the ground. Using your hand carefully spin the wheel 4 or 5 full turns. You are feeling to see if it gets harder to spin at certain places as it turns. You are also listening to the noise the wheel makes as it turns. Listen for a quiet gritty sound. A good condition brake disc will give a steady, quiet, gritty noise all the way around as it spins. A warped disc will be an 'On then Off' sound. Use the comment button below with more information and I'll see if I can help you more.

that is the engine misfiring.
get a tune up, at 60k miles. every.
open hood see engine shake? that is called misfire.
and wastes fuel.
and will get progressively worse utill stalled.

need a little more detail here.
Rumble from front? rear? left or right side?
And shake in the steering wheel?
How big a bump and how fast were you going?
Was the rumble instant after the bump?
Is it constantly making the noise or now and again?

if you let off the gas and it stops vibrating, have the cv axles checked. if the steering wheel shakes at parking lot speeds, a tire is bad.

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Your disc brake rotors are warped. Solution? Remove the disc brake rotors and have them turned on a brake lathe. Warpage comes from a hot disc brake rotor going through a puddle of water which causes the warpage. If you have a Parking Brake, get in a safe area and when stopping, ONLY apply the parking brake. If the steering wheel stops shaking...there's your problem...warped disc brake rotors.

Sounds like a bad wheel bearing to me. Jack up each corner of the car one at a time and grab the tire. Shake like crazy on it - if it moves, you've found a bad bearing. It should be solidly on the hub with no play whatsoever.

Try turning the steering wheel (hard) right and left while turning the key. Steering lock , and ignition lock work in conjunction with each other.

when slowing to make right turn is it more like a clicking noise, does it do it while turning or just when slowing down. If it is while you are turning it is most likely the right cv axle drive shaft popping. When at a stop with ac on does it shake if you put it in park and how do the rpms read in park or in drive. Did they change the trans mount also dont thank that is it just trying to get a better understanding so I can better assist you.

If the vehicle is FWD then you may have a CV outter joint going out and/or possibly a wheel bearing failure. I would have this checked out by a professional shop AS SOON AS POSSIBLE. It also could be causing the power steering alert to come on because it is detecting the vibration in the wheel arm for the steering. Good Luck, RAC