Steering wheel shakes at times when accelerating and when applying brakes, noise coming from rear wheels

If when applying the brakes the steering wheel shakes then your front rotors are warped and at the least need resurfaced and most likely the brake pads replaced. But more likely the rotors are too thin which has allowed them to overheat and warp which causes the shaking. In this case new rotors and pads are in order. If when braking tou feel the shaking in the seat of your pants then most likely the rear brakes are the issue. Same problem as the front, warpage, but your rear brakes are most likely drums. Again, possible resurfacing and new shoes; but more likely new drums and brake shoes.

But I wonder? Did this vibrating just start all of a sudden or maybe after a near accident requiring you to brake very heavily and possibly overheat the brakes. Or my main concern, to keep this from happening again, after pulling some type of trailer that you intend on pulling again. Pulling a trailer without its own brakes puts a much heavier load on your trucks brakes and just repairing them won't keep it from happening again if you ARE pulling a trailer. Higher quality rotors/pads/drums/ and brake shoes can help but if the trailer is too heavy for the truck nothing but installing brakes on the trailer or a lighter trailer will stop you from warping them again.

Lastly, in general when you let off the gas and the whole vehicle shakes the driveshaft/u-joints are common causes. Same with the steering wheel if it shakes when letting off the gas, but only on a 4x4 or AWD vehicle. So check your driveshafts and u-joints and make sure they are tight. But unless you find something obvious there I would wait until you have the brakes sorted out and see if that doesn't stop all the shaking.

Good luck!

bad wheel bearing/ faulty brakes

sounds to be a bad wheel bearing

JUST steering wheel or entire shaft, and or dash? do u feel it in seat? trying to distinguish EXACTLY what parts are shaking is very important

bad tire, gad wheel bearings , loose steering tie rod ball joints , wheel alignment
take it to a front end specialist for a full inspection

If it only shakes when accelerating and not when coasting or up to speed it is probably your inner c.v joints.

If it shakes the whole time it is most probably one of the following

-Wheel balancing
-damaged tyre
-bent or buckeled rim

It sounds like the rear wheel is out of alignment, or loose. Could be a worn bearing. Torque is applied to the rear wheel as you accelerate, and lost when left off. If I am not mistaken, the rearend is set with a dyno machine at the dealer. Try putting the bike on a lift to get the rear wheel off the ground. Grip the rear wheel on opposite sides with both hands and lightly alternate pull right and left, and top and bottom. If you get any movement from the wheel, it is time for service as this is an indication of the bearings wearing out. If you recently had the rear end serviced, they may not have gotten everything tightened properly, or misaligned the rear wheel.

If I understand correctly the steering wheel must be held in different positions to keep the car going straight while accelerating/braking accelerate hold wheel to one side ...braking hold wheel to other side. check strut mount(rubber mount on top of strut where it contacts car body)

check your upper control arms, does it happen when you step on the brakes or just when driving, you might need a balance of all your wheels or the tires you have could be bad.

I believe you need your disk brake rotors turned to be true again or have them replaced. Whe you use your brakes a lot and kit puddles of cold water then it will warp the rotors which will cause the violent shaking. You may not feel it in the brake pedal even if it is warped rotors. If you don't think you have warped rotors then you should be very careful driving it because it will probably be in the steering sector arm or the idler arm on the steering shaft.
Dale