Home vibration is a non-issue for solid state stuff. Part of my job is directly involved with doing vibration testing, and the amount of vibration a home headphone setup sees would be so low as to be barely measurable, let alone cause any issues that the mass of a power cord would affect. And even assuming there is a level of vibration that would cause any measurable effect, the mass of a power cord would do absolutely nothing to dampen resonance nodes and the like inside the amplifier case and among the components inside the case. It's more arguable that an amplifier in a home theater setup - where it will see much, much more low-frequency noise and vibration - could have a slight chance of causing issues with a solid state amp, but with headphones, no.
As far as power connections (to be distinguished from source cable connections), it's either good or its not. You're either transferring a stable waveform or you're not. Modern transformers are capable of dealing with minute voltage differences quite capably with no untoward effects - hell, depending on the specification, to achieve a CE mark, a product must operate safely at +/-10% of its rated input power. Unless you're seeing massive voltage swings - which a power cable would neither cause nor remedy - a connection that transfers AC is a connection that transfers AC. It can be unobtanium-plated gold connectors, but a stable voltage is a stable voltage. If there's a loose or bad connection, the issues will be on a much higher scale than, "I think this cable has a slightly reduced soundstage and a tiny bit of treble roll-off." It would be easy to measure that sort of thing, too - run everything through a power meter, use Cable 1 and note the current draw, then use Cable 2 and note the current draw. A bad connection would result in lower voltage, which would result in higher current draw, and once we establish a normalized delta for current draw variance using one power cable, it would be a simple exercise to compare a different cable. Assuming audio amplifier boards have test points, one could also measure the voltage during operation with different power cords and note any changes there.
I think, if power cables make any difference at all, it's much more to do with real, measurable electrical phenomena - capacitance, inductance, and the like. Undersize a power cord and you're create a low-voltage/high current situation that could potentially cause issues with components working. Too large a power cord, and you can likewise run into problems. But all of this would be measurable, and so far, I haven't seen anything even resembling actual measurements by power cable proponents. (Yes, I know, sound science vs DBT free, so I'm not going to pursue that point of the discussion any further - simply pointing it out).