Intermittent shaking while driving and braking

It was a caliper after all. Thanks for all the help.

My car, Toyota Corolla 1997, started to shake two months ago, when I put it to reverse gear to go out of my parking place. It continued to shake while I was driving, though not as strong as at the beginning. I went to mechanic, he replaced three serpentine belts and the timing belt in the engine and problem disappeared. Two weeks ago car started to shake again, but after 15-30 minutes of driving the vibration stopped and I didn't notice it for several days. Then shaking started and stopped again, as first time. I went to mechanic, he replaced all four engine mounts - front, rear and sides. Since that car shaking is even worse. Right now my car shakes pretty strong when I stop on the lights. When I drive it vibrates a little bit, but not as bad as when I stop and press the brake pedal. One of the mechanic hypotheses was that the compression at the engine cylinders is different. It cost me 80$ to check this hypothesis and to confirm that it was wrong - the pressure was the same in all cylinders (190). There are other hypothesis, which are more expensive, and it would cost me the price of my car to check all of them.
I will be really appreciative if anyone who had similar experience with the happy end can give me advice regarding my case,
Thank you very much,
Natalya

Hi, the check engine light was on when the mechanic gave car back to me (he replaced mounts). I told him about that problem and he turned light off. The engine is on place (maybe I didn't understand your question - what does it mean “is the engine missing?”). I have a feeling that car behave differently than before, maybe a little slower, some new sound, though I am not sure.

The engine light is off and the compression was measured two days ago, it was 190 in all four cylinders. As to speed, I am not sure that the car is slower, maybe a little bit, maybe not, hard to say now. The main problem is that car is shaking when I press brake pedal and stop.

The car shakes when it is not moving and I press on brake, for example on stop signs or on lights. If I press on brake, but car hasn’t stopped yet, it doesn't shake. Also, it shakes (less) in idle conditions. The brakes have been checked recently, rear brake shoes were replaced, front ones were OK, brakes fluid was changed.