I have a quiet system with a PicoPSU 150 and there's this little tab on the back of the AC jack. Should I attach some kind of ground wire there?

I ask because I hear annoying system noises through the speakers, staticy buzzing that seems to correspond to PC activity and therefore current draw. I thought it was my cheap sound card (the onboard one sounds even worse) so I got a $20 Creative USB sound adapter to move the sound generation to outside the case, but the noise persists even when I use a 6' USB cable.

I've tried HDMI and that works well - for a while. After a few weeks I get this problem where there are very loud crackling sounds whenever any audio is playing. But before that it's crystal clear.

The problems you are describing can be an earthing fault, but it's more likely that it is just a bad audio codec or some other problem. By connecting your audio jack to another device, you are effectively grounding the signal path between the two. Also, if you really think it is a grounding problem, the audio defects should not occur when using headphones.

A common cause of crackling noises is interference from the CPU VRM, especially when the CPU is near-idle and switching in and out of low-power C-states. This can be verified by disabling C1E mode in the BIOS.

The tab on the picoPSU jack is not a grounding wire. Most power adapters have two-prong (8-shape) or otherwise ungrounded connections, so grounding is essentially impossible there. If you happen to have a power adapter with a grounded (three-prong, mickey mouse) connection, you can ground your chassis by connecting it to the negative supply terminal from the picoPSU. However, before you do this make sure you double-check that the ground connection is indeed connected to mains earth, and double check afterwards that you have done it correctly.

It's a picopsu 150W and the brick does have a ground prong.

For now I've switched back to the HDMI connection which is free of this problem, but it's only a matter of time before that starts spazzing out on me, as has happened twice before.