I plugged in my expensive audiophile headphones and didn't hear anything.
I plugged in my ipod earphones and didn't hear anything.
I plugged in some shitty MPIO earphones and heard a slight electronic hum.

I would say it's cause of your headphones.

 

Well not so sound rude or anything, but you can't be right about that.
I'm an audio engineer so I know my gear, these are a pair of Sennheiser HD200 headphones, among a collection of other Sennheisers.
I don't have any problem with these whatsoever in any other application.

edit: I've come to the following conclusion -
by turning off the audio completely, I still have this issue, so it's related to starting up the wifi-function. wifi drains more power, thus an electronic hum

I wouldn't be surprised if this is the case with all samples of the console
I'll see if I can get a nice recording of the noise for you to listen

 

Absolutely right, and it´s a shame ´cause some of the "Surround" effects in some games are really good.

 

I hear it too, I use Koss Sparkplugs (very, very nice headphones for the price) and get a very noticeable electronic humming (always the same level no matter the volume) when I change screen brightness, load an image, play a game, when icons are shifted, and especially while loading ROMs. I have noticed it on my sister's G6/DS also, it's not really bad. You just need to turn the volume up to drown it out, Nintendo didn't need to worry about Audio all that much since it was designed for gaming, and gaming only (since the system has so very little RAM I don't think Nintendo really worried about the "quality" of the system, they just cared if the games were fun with what was at do).