To certify any Commercial Jet Plane worthy of flying,
a 2 engine plane must be able fly with just 1 engine running
a 3 engine jet like the Lockheed L1011 or DC10 with just 1
a 4 engine jet with just 2. Although not a requirement, the 747, A380 could fly on one engine but are not be able to maintain the cruise altitude indefinitely and will eventually become long-range gliders so they must land immediately.
The landing approach and procedures vary on how much the plane weighs prior to landing.
Would the passengers be notified, asking to take the brace position during landing?
Brace Positions are only assumed on impact emergencies, like crash landings. Losing an engine doesn't mean you are going to crash so no brace will be assumed.
Would such event have an accident investigation report?
There will be an investigation but if the plane landed safely it wont be filed as an accident. The plane plus the engines passed its FAA, EASA, etc Certification so it may be a number of reasons why the engine malfunctioned, like bad maintenance, faulty parts etc.
This is an extract from wikepedia -- http://en.wikipedia.org/wiki/Turbine_engine_failure
Shut downs that are not engine failures
Most in-flight shutdowns are harmless and likely to go unnoticed by
passengers. For example, it may be prudent for the flight crew to shut
down an engine and perform a precautionary landing in the event of a
low oil pressure or high oil temperature warning in the cockpit.
However, passengers may become quite alarmed by other engine events
such as a compressor surge — a malfunction that is typified by loud
bangs and even flames from the engine’s inlet and tailpipe. A
compressor surge is a disruption of the airflow through a gas turbine
engine that can be caused by engine deterioration, a crosswind over
the engine’s inlet, ingestion of foreign material, or an internal
component failure such as a broken blade. While this situation can be
alarming, the condition is momentary and not dangerous.
Other events such as a fuel control fault can result in excess fuel in the engine’s
combustor. This additional fuel can result in flames extending from
the engine’s exhaust pipe. As alarming as this would appear, at no
time is the engine itself actually on fire. Also, the failure of
certain components in the engine may result in a release of oil into
bleed air that can cause an odor or oily mist in the cabin. This is
known as a fume event. The dangers of fume events are the subject of
debate in both aviation and medicine.
Would this report later be available for general public, as are the reports of various significant air disasters?
It depends, as you read above the flight deck crew may shut down an engine because of low oil pressure. Would that have significance of becoming public knowledge?
However as an example, if the problem of an engine failure escalates to the manufacturer then the airline may ground the fleet and that then becomes public knowledge and the reason why because the airliner would give media statements.
It all depends on the severity of the problem. Like this one that got reported --- http://www.b737.org.uk/accident_reports.htm
20 August 2007; B-18616, 737-800, 30175/1182, FF 11 Jul 02, Okinawa,
Engine #1 caught fire shortly after docking destroying the aircraft.
No fatalities. The cause was the detachment of a slat pylon bolt which
fell off and pierced the fuel tank which leaked and ignited. According
to the JTSB the washer that should have held the nut in place probably
detached during maintenance 6 weeks before the incident. The design of
the nut has since been changed to limit the likelihood of detachment.