Lots of great posts and useful information here.
I’m going to approach this more from the "what" perspective and not so much the "why" aspect. I’d like to point out that I come from the "reverse bias" perspective, whereby, the concept of premium cables pegged the needle on my "BS Meter". 60 hertz is 60 hertz. It doesn’t change. Electrons travel at the speed of light. And, plugging in something that pulls 3 amps into a (factory supplied) power cord rated at 10 amps is well within the current demands of the cable. I was insulted when our cable rep came around and offered to audition power cables with me. I, unapologetically, sent him and his high-priced cables packing. Until the day he asked me "as a personal favor" to take some home and listen to them. When the evening’s listening session was completed, I didn’t just drink the Kool-Aide. I chugged the whole pitcher.
My take on power cords is summed up in one word: AGILITY. Quicker, faster, more energetic power delivery. This is the "what" I referenced earlier. When musical information is presented to the power supply, it reaches back to the wall outlet to get more juice. If the power delivery is slow and/or anemic, the musical elements are there and gone before the power supply says: "’Hey, I’ve got a job to do here!" Microdynamics are undefined ripples vs clear swings in energy that reveal vibrato, trailing voices and instruments, and subtleties. Demands for transients are met with fully defined elements providing rich dynamic contrast. Those "POWS" are "POWS" and not "smushes". Ah, then there’s the improvements in focus and detail. Those formerly hidden spacial queues tell our brains where things are positioned in space a lot more accurately. And, more agility reduces strain and glare.
In, literally, hundreds of cases I have not heard a situation where power cord upgrades did not improve the sound of whatever they were plugged into. From price-driven mass merchant low-fi gear to "reference"systems it was unmistakably better. So, I struggle with those who have had negative results with premium cables. Admittedly, I only have experience with power systems in our region in a "typical" home with equipment we’ve had exposure to. So, I’m not saying that there are conditions where power cord upgrades don’t matter (source, low current, high current, integrated systems, separates, etc). I just haven’t experience one. Yet.