Okay, I've really dug into this. To sum it up:
MPC-HomeCinema and it's SHOUTcast Source Filter are the culprit. And to make matters worse; I completely overlooked the DC-Bass Source Filter its shoutcast support out of the box.
After quite some testing this filter passes with flying colours for all sorts of audio streams, whereas the MPC-HC SHOUTcast Source Filter fails miserably.
It's because I abandoned Zoom Player (where the DC-Bass Source Filter works perfectly*) some years ago in favour of Foobar that I thought there's something wrong the filter's shoutcast support, but it seems it's in fact MPC-HC which is the culprit. Somehow it can't call forth the DC-Bass Source Filter for audio streams.
If you'd like to see the full test report, click here
Overlooking everything now, I think with the next update I'll replace the MPC-HC SHOUTcast Source Filter in favour of the DC-Bass Source Filter. Why? Although they might be able to fix the issues with MP3 and AAC, I don't know if we can expect Vorbis support and the DC-Bass Source Filter has already just that.
If you're using MPC-HC, you can forget streaming, at least for now. To listen to online radiostations I suggest you use Zoom Player, or if you're desperate MONOGRAM GraphStudio
(GraphEdit is old!), together with the DC-Bass Source Filter. Be sure to unregister the MPC-HC SHOUTcast Source Filter first!
* Only OGG-streams are a bit troublesome. They won't play at the moment, but I think it's an issue with Zoom Player rather than with the filter. With MONOGRAM GraphStudio OGG-streams are working fine.