It's a VERY common problem. From my experience, in fact, it's the absence of noise that is a very rare exception !

Perhaps I am more tolerant than some people. The noise I was referring to was jarring and unmistakeable, and completely unacceptable. My current set up makes a barely audible "speaker hiss" if you put your ear right next to the speaker cone at listening volume, but from a meter or two away it's completely inaudible, and that's good enough for me.

When you run your laptop on battery, usually it switches from a "high performance" mode to a more "battery saving" mode, and that might be the cause of drop-offs. Just make sure you're running in "high performance" mode but, of course, that drains a lot of power from the battery.
Regarding the problem you describe, there is another solution: cut or better remove the ground cable of the PSU. I know it isn't considered as "safe", and in theory you shoudn't do that but it's the easiest way you get rid of those pesky noise. Just be cautious. I did it on 5 laptops already and their audio outputs are all completely noise free.

I can see that that would work, but fortunately just substituting a better quality power supply got rid of the noise for me. I see there's a further suggestion in response to this. I suspect this actually is a more common problem than people may realize.

If you get audio hum no matter what you do, unplug the power supply. If you don't get it with your laptop running on battery, that's the problem. But I found that running PTQ on battery has its own problems (weird occasional drop-offs, and even running the controller with its own power, the battery barely lasted an hour and a half). So if this is your problem, I suggest you replace the power supply cord/transformer with a better shielded model. More or less in this respect you get what you pay for, so you can probably find something that will not do this.