Electronic – Why aren’t there any 3-phase domestic power-outlets, or any interest in 3-phase for household (mains) power or anything relatively low-power

Cost.

• A three-phase distribution system would require four wires to each house - the three phases and a neutral connection. Single phase only requires two.
• A three-phase fuseboard would be required.
• Four wires plus earth would be required in the domestic wiring.
• Five-pole connectors would be required.

Figure 1. 3-phase connectors - European and North American.

... and it appears that the only 3-phase power outlets in existence are meant for ultra-high-power industrial equipment.

It's usually worthwhile above a couple of kW if three-phase power is available, as it is in most industrial and commercial buildings.

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.

This is true.

3-phase would naturally make less power-supply noise, so we could have cleaner power with smaller capacitors.

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.

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.

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.

I make guitar pedals as a hobby, and power supply noise is quite a nuisance.

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! ;^)

So why is it that home power outlets aren't 3-phase anywhere in the world, or even interest in it?

Still cost.