I have been hearing loud screeches every now and then from my car engine.
Sometimes it screeches when I start up and sometimes momentarily when I accelerate. Is the engine bearing about to fail?
The annoying noise is not anything as serious as failing engine bearings.
The screech you hear is from a tired accessory belt, which is either the one that drives the alternator and fuel pump (and the power-steering pump on many cars), or the air-conditioner compressor.
These belts are subject to wear and must be replaced periodically. Old belts become brittle and lose their grip on the drive pulley.
When they slip, the rubber sliding on the steel pulley lets out the loud screech.
If you continue driving with the same belt, it will eventually break. This could result in overheating because the water pump is not operating, or a discharging battery because the alternator is no longer turning.
If it is the power steering belt, your car's steering will become heavy. If it is the air-conditioner compressor, you will not have any cool air blowing out of the vents.
It would be advisable to change the belts as soon as possible. It is a relatively inexpensive job.