It's yet another area of hi-fi where just about everything said about it is bollocks (like several other areas!) propagated by people who know nothing about electronics. If people knew more about how things actually work they would understand how they may be affected, or not affected, by external influences. Unfortunately there is a vocal minority whom, inspite of repeatedly saying "I know nothing about electronics BUT...." then go on to talk total tosh. These people apparently receive a "special" education in electronics from hi-fi journalists, sales people and "Steve of the so and so forum" which contains many chapters completely unknown to us Electronic Engineers....
Bad, distorted mains could well affect a turntables synchronous motor adversely if it is powered directly from the mains and not from a "Valhalla type" mains regenerator. This is because that mains waveform is directly driving the AC motor and so the distortion etc on it will cause greater vibration and torque irregularity. On the other hand mains so bad as to cause a transformer to audibly hum and vibrate will likely have no affect on the sound of an amp being powered by said transformer (assume that the audible hum is not bad enough to spoil things!).
Other than the case of an AC motor being powered directly from the mains it's hard to think of much else that's likely to be affected by bad mains... Crude or vintage valve gear which has heaters AC powered from a 6.3V winding on the mains transformer may show an increased noise floor...
All this is referring to standard mains transformer supplies. A switch mode supply will be even less affected in most cases.... but can of course generate copious interference. This is nothing to do with the mains itself.
