Visa requirements: South African passport, Danish residency

Capie, why don't you just make an inquiry and tell us what the answer was.

Here is what wikipedia says about this subject. I know that this is not an official source so you should confirm. However, if it is correct, it implies that you can travel for up to 90 days if you have a national visa issued by a Schengen state.

Right of stay
A third-country national who has been granted entry may stay in the Schengen area and travel between Schengen states as long as the conditions for entry are still fulfilled.[28] For stays which exceed three months, so-called national visa (category D) are issued by the relevant Schengen state where the third-country national intends to reside. Any third-country national who is a holder of a residence permit of a Schengen state, which is granted for a stay which exceeds three months, is allowed to travel to any other member state for a period of up to three months.[29]

It took me long enough, but finally I got round to asking Danish Immigration, who told me that the Danish residency permit does allow me to enter the other Schengen countries. Phew!

Stockholm28 gave you that info on March 4. LOL