What are the advantages of using a three phase electricity service compared to a single phase service?

Answer for countries in Europe and other world areas running a 50 Hz supply service.

Basically, the higher voltage available in a 3 phase service (415V), allows much greater power to be supplied for a given conductor (cable) size. Each phase is at 230V to earth/neutral and 415V between phases. Most single phase standard domestic installations are rated at 100 amps, meaning that the total power available is 100A x 230V = 23,000 watts. (23 kilowatts)

Using the same size cable as a 100A domestic installation uses for its service lines, a three phase installation makes much more power available to the site's load equipment (compared to a single phase service) because there are three service lines, each rated at 100A, making a total of 300A, all at the higher voltage, meaning that the total power available is 300A x 415V = 124,500 watts. (124.5 kW)

This amount of power is generally used in industrial and/or commercial installations, or perhaps domestic installations where a large pottery kiln, or some other heavy load is connected.

It should be noted that the current carrying capacity of wire is the same for all voltages and phases. Number 10 wire is good for 30 amps on a single phase system and good for 30 amps on a three phase system. Ampacity does not increase just because it is a three phase system.