In my experience it is not "common" for air carrier aircraft to taxi with an engine shut down. It does happen of course, but there is generally a specific reason behind that procedure if used.
Reasons may be:
Fuel load is close to the minimum required for the flight at the start of taxi (a variety of reasons for this).
Long line up on the taxiway awaiting takeoff clearance.
Captain discretion to save fuel.
For most Air Carriers, the engines are started using air pressure from an APU (auxiliary power unit) affixed to the fuselage (at the tail of the aircraft). The start procedures are initiated while the aircraft is being pushed back from the gate by a tug. Shortly after the aircraft is pushed back and generally aligned for taxi both engines (twin engine aircraft in this example) have been started and an "after start" (or similar) checklist has been completed. Besides the engine noise, which is sometimes quite low and hard to hear, the end of the start procedure can be noticed when the lights blink in the cabin as the electrical power generated by the APU is being replaced by the engine driven generators. Also, there is generally an increased airflow into the cabin as air from the engines is vented to the cabin by the air conditioning system. Very little air is available during engine start from the APU because the air is being used to start the engines.
Anyway, this is probably a more expansive explanation than necessary for your question (I just wanted to give some context to the process of engine start for air carrier jet aircraft).
If the aircraft is taxied out using a single engine, the crew can either start the remaining engine at some point prior to takeoff using the APU (if left operating) or by cross-bleeding the engine air from the running engine to the engine now being started.
If the aircraft is taxied with one of its engines not started at the gate, the crew has to run an additional checklist during the taxi as and after the engine is started. Not a big deal, but a bit more workload.
Lastly, there have been occasions where problems have resulted from taxiing with one of the engines not started. For example, many years ago a B727 (3 engine aircraft) started its takeoff roll with only two engines running because the crew failed to start the third engine. It crashed on takeoff. Also, there have been occasions where the aircraft was not configured properly for take off (e.g., engine bleed switches in the wrong position) due to a failure to properly follow/run the appropriate checklist for starting an engine during taxi or configuring the bleed/pressurization system properly for take off.
Again, taxiing with an engine not yet started does happen. But considering how many flights occur every day in the world it would be difficult to say this procedure would be anything close to common.
Just my two-cents