In relation to the shutting down of one engine on a three engined jet during cruise, eg. Tristar

Is this common during all flights or in certain circumstances?

Why is this done?

Only in an emergency.
Even in military use we didn't shut down one in the cruise. The same is not true for the 4 engined Nimrod or P3 where engines are shut down to save fuel.

For Civil Operations - Only in circumstances of emergency, engine failure, or other abnormal condition requiring the engine to be shut down. NEVER as a normal procedure.

As stated, military operators of patrol aircraft may do this to increase range and/or endurance.

To be able to do such, you would have to be in such a position that in the event of the failure of one of the remaining 2 engines, the aircraft could still fly on one engine - this would be highly doubtful with any reasonable payload in such an aircraft.

The Nimrod at loitering level above the sea has one engine shutdown and one throttled back to idle, while using the remaining 2 to stay aloft - this would enable a situation such that in the event a failure the idle engine could be spooled up to attain that extra thrust, but in normal circumstances fuel is saved, increasing endurance and/or range.