Which standard is applicable to the electric power system in civilian US aircraft?
The answer is not straightforward.  Power supplies used in commercial aircraft LRUs are required to meet RTCA DO-160G, sections: 16 Power Input, 17 Voltage Spike, and 18 Audio Frequency Conducted Susceptibility - Power Inputs.  

The issue is this is what the consumers of aircraft power have to deal with.  It doesn't actually specify what the aircraft power distribution system has to provide. Though by inference, the power generation system must deliver power within the bounds of what the consumers can accept.  

In my experience, aircraft OEMs design their power distribution system and then levy specific requirements on the power supply design of all consumers on the aircraft. While DO-160 covers the majority of the need, the OEM requirements will include additional requirements deemed necessary by their design and analysis. As a Boeing engineer explained to me, "The requirements are not to protect your power supply.  They're there to protect my power bus."  

The reality is aircraft power is a small dynamic system and it is impacted by everything on the bus.  It has to be designed as a system unique to each aircraft. Since it's part of the aircraft, the aircraft rules come into play.  For air transport aircraft, that means 14 CFR Part 25. The applicable portions include 14 CFR 25.1310 and the Electrical Systems and Equipment subject section.  

Additional guidance is available in Advisory Circular AC 25.1353-1a ELECTRICAL EQUIPMENT
AND INSTALLATIONS.