There is a list on Wikipedia.
Radar guns use an optical Doppler effect to measure speed. Their acoustic equivalent is used in medicine, where it's called Doppler ultrasound and used to measure blood flow or other sorts of motion in the body.
Animals that use echolocation can use the Doppler shift to gain information about the motion of their surroundings.
A sonic boom occurs when the Doppler shift shifts a frequency to infinity.
I guess one can continue to concoct scenarios. I wonder whether, when you drop a cat off a cliff, you can hear the pitch of its screaming drop as it accelerates. (It's not all that cruel - cats can usually survive a fall at terminal velocity.)
You could use it to determine which way a whale is swimming if you have two boats, both listening to the same whalesong. You could even use the Doppler shift to gain information about the position of the whale because both the whale's position and its velocity contribute to the observed Doppler shift at any given place. (I don't have any information about this actually being done, but it might be an interesting problem to work out the locus of possible whale locations for an observed Doppler shift.)
I was also curious about whether the Doppler effect gives us information on the motion of the crust that moves during an earthquake. I found this reference which suggests it does.
Do pilots of a supersonic jet hear the sound of the engines when it breaks the sound barrier?
Yes, but not through the air. The sound is largely generated internally to the fuselage of the plane. While sitting on the deck ready for takeoff, then the engine would likely be much louder but the cockpits are often surrounded by heavy materials reducing sound transmission anyways. It's just that fighter jet engines are so obnoxiously loud, e.g., see notes about this at: https://physics.stackexchange.com/a/281767/59023.
However, I think that sounds originating from in front could be heard with a large doppler shift causing the sound frequency to increase. Am I correct? Or does the shock wave in front of the jet somehow totally destroy the sound waves?
Your second questions is likely closer to the truth than the former. The reason being that shock waves cause an abrupt change in the index of refraction for light and for compressive sound waves in a medium. So much so that sound waves can actually reflect off of the shock wave (e.g., sometimes shock waves reflect off of each other). Most linear sound waves that start outside of and in front of a supersonic jet would likely be reflected and destroyed. That is, immediately after reflection the wave would be overtaken by the shock since the sound wave propagates only at the speed of sound whereas the shock speed is defined by the piston/driver (i.e., supersonic jet here) speed.
Regardless of this, one would need a rather intense sound to actually be heard. That is, suppose we ignore the issues with crossing the shock barrier. The sound would need a greater intensity than that of the jet, which is not insignificant as I noted above. Further, the sound would need to be extremely loud to be heard inside the pilots helmet which is intentionally designed to reduce outside noises for communication purposes. My guess is that something like a missle explosion may be heard, but unfortunately it may have to be close enough to cause damage to the craft.
I will pester a colleague of mine that used to be an air force pilot to get a more definitive answer.
Update
I spoke with my air force buddy and he said a few things that kind of confirmed what I had originally thought. So most of the noise in the cockpit is due to the engines and it is primarily transmitted through the solid body of the fuselage. The cockpit is intentionally designed to reduce noise from outside sources. Further, the helmet has additional noise reduction methods implemented to allow the pilot to mostly hear comms and not much else (besides the vibrations of the engine, of course).
In general, shock waves reflect off of each other so explosions ahead of a supersonic jet might be felt but likely not heard unless they are extreme (in which case the integrity of the jet may be more of a concern than whether the pilot can hear the external noise source).
Best Answer
Because in order to travel at supersonic speeds, human beings must be enclosed in a rigid metal tube of some sort. Also, these metal tubes they ride in at those speeds generally tend to be insulated against noise from the outside.
As for trying to place some sort of microphone outside said metal tube, the propulsion system would risk drowning out any atmospherically transmitted noise (i.e. noise can be transmitted through the body of the structure).
Now, if you run the math, it still doesn't work quite like hearing it backwards (see the comment by eudoxos). Although you would encounter the soundwaves in "reverse" order, the shockwaves around you would disrupt anything around you as to make the notion of noise from them irrelevant.