Rear brake have brakes but a burning smell then a big puff of blue smoke came from one rear tire,bad grinding noise

It's the car pulling to that side? Sticking brake pad possible. When did you have brakes done last off at all? Front brakes go out twice as fast as rear brakes. If you can Charlie jack up that side put car in park, set emergency brake and chock rear tires. Pull off that tire and see if rotor is grooved or scored badly. It may have been sticking and wore down faster. Also check CV BOOT. If torn u joints get dirty and will grind. Hope this was some help.

Your description makes me worry. If you have a strange, unknown screeching noise coming from the right rear wheel and now there is a burning smell. Please stop using it and have it looked at as soon as possible. There are a few things that could be the cause of these symptoms. Wheel bearing failure, seized brake caliper or frozen caliper sliders, damaged brake line, seized E-brake cable, or improperly installed brake-pads. In any case, if you continue to drive this car, with these symptoms. You may cause a break-down (requiring a tow), or cause a fire. Let me know if I can answer any other questions or how the repair goes, good luck.

Having this happen on both front brakes makes me think you have a bad master cylinder. What work have you done to it prior to this problem?

Yes the white smoke from the rear at the wheel means that the breaks need to be changed/replaced.

Could be brakes pads/shoes are worn out, brake drum/rotor grinding from metal to metal contact. Or a bad brake caliper. Parking brake could be on. Or rear wheel bearings.

could be the rear of the intake manifold leaking

ok blue smoke is you valve seals in your head.the shaking could be wheel bearing need to check them

is the noise is constant when you are driving is a bearing , is the noise is when you brake, check the brakes..jack the car and spin the tire you will here the noise ..hope this was very helpful..

Try to jack up the wheels and turn them to see if there is no seized brakes/calipers. Let us know how you get on.
Cheers