Since I discovered what 3-phase electrical power is on Wikipedia, I have noticed that it's associated with very high-power applications. I have not been able to find anything on the subject of 3-phase power for anything but very high-power applications, and it appears that the only 3-phase power outlets in existence are meant for ultra-high-power industrial equipment. I find this strange because I imagine (lower voltage) 3-phase would be better than single-phase for household power outlets.
The main reason that I think 3-phase would be better, is that it could be rectified into cleaner DC power. 3-phase consistently provides power while single-phase repeatedly goes down to 0 volts before rising back up. 3-phase would naturally make less power-supply noise, so we could have cleaner power with smaller capacitors.
I make guitar pedals as a hobby, and power supply noise is quite a nuisance.
It might possibly be safer, as the AC voltage could be lower (in case if two wires are shorted together by a person), but it could still have a higher DC output power with a 3-phase rectifier, and possibly other things.
So why is it that home power outlets aren't 3-phase anywhere in the world, or even interest in it?
Figure 1. 3-phase connectors - European and North American.
It's usually worthwhile above a couple of kW if three-phase power is available, as it is in most industrial and commercial buildings.
This is true.
This may not be true. In single-phase and 3-phase the diodes only conduct close to the peak of the voltage waveform when the incoming voltage exceeds the capacitor voltage. This results in pulses of current on each phase and causes odd harmonics on the current waveform.
Figure 2. A VFD drive showing the rectifiers, a filter to reduce noise, the six switching transistors to PWM the current to the 3-phase motor. Source: ECMWeb.
Variable frequency drives (VFD) use that principle to rectify the mains and generate a high-voltage DC supply internally. As you stated, the capacitor value can be much smaller when using a 3-phase supply than it could with a single-phase supply. VFDs are commonly available in the range 1 kW to hundreds of kW.
That sounds attractive but the advantage of higher voltage is that currents are reduced and voltage drops and power losses along the wires are reduced. In addition, we still need some high-power loads such as water heaters in immersion heaters, washing machines and dishwashers.
Figure 3. A 3-phase transformer.
Not as much of a nuisance as requiring a three-phase transformer. I don't think anyone makes 3-phase wall-warts! ;^)