In the US, three phase power seems to be completely unavailable in rural areas, except in small pockets of industry.

So how do the power companies run electricity to those areas? They don't use DC, do they? Or do you mean individual houses are typically wired with just one of the phases?

US residential and rural areas typically have 120/240v split phase (single phase) 3-wire service, (2) 120v hots and a shared neutral. Commercial customers can usually get three-phase power.

Correct. It can be prohibitively expensive to have 3-phase power run to your house. Most consumers who need it, e.g., hobbyists, will use a phase converter.

The only person I know who actually had 3-phase power run to his house was using it for a ceramic kiln.

Doesn't a farm count as a small pocket of industry?

(Writing from Denmark, where my small apartment has three phase power. It's standard, I don't know why.)

Living on a rural US farm... In our area, 3 phase was not ran to our farm until we paid a large lump sum for the utility to run it from the nearest substation to us. Before that we only had single phase.