Visa requirements: South African passport, Danish residency

Your Danish residence permit applies only to Denmark. It is of no signifigance at all in any other country. All normal rules apply which means you can visit Schengen countries for 90 in every 180 days like any other tourist. As a South African you are required to apply for a Schengen visa.

But having a Danish residence permit may help in getting the Schengen visa.

But the guy already has a Danish visa and is living in Denmark. What would he need another visa for? There are no border controls anyway.

#3 - in order not to be illegally in whichever country he travels to. There may be random border checks, or other checks, he might get into an accident etc. - I can think of tons of situations where he may have to show his passport.
A residence permit in one Shengen country doesn't automatically grant you access to other Schengen coutries (residence permits are usually national visas). What he needs is the 90-day Schengen tourist visa which will allow him to travel in all member countries.

Which part of my response #1 did you find hard to understand iluvmangos? The OP needs a Schengen visa to legally visit any Schengen country other than Denmark.

With the caveats that the rules may have changed, and circumstances may vary between countries: I am a Philippine citizen and used to live in Spain with a Spanish residence permit. That allowed me to visit, but not live in, any other Schengen country without getting a new visa. I visited Italy, France, Iceland, and the Netherlands without a visa- I just showed my Spanish residence card when asked (and I was asked several times); no hassles. I did have to get a separate visa to visit the UK.

With respect maaraw, that no one stopped you for not having a Schengen Visa does not mean that legally you didn't need one to legally be in a Schengen country. They simply chose to ignore the situation based on your residency card. All immigration officers have discretionary power. Any one of them could have stopped you and denied you entry into his country, they just chose not to bother with the hassle. In other words, you must have come across as innocent enough with no ulterior motives suspected. Congratulations.

A visa of any kind issued by a specific country is valid only in that specific country. That isn't hard to understand surely.

The Schengen rules re nationalities that require a visa are straightforward and simple to understand. A S. African needs one to visit ANY Schengen country. Surely that isn't hard to understand either.

No country can give you a residence card valid in their country, that supersedes the requirement for all other Schengen countries. That shouldn't be hard to understand either given the previous two statements. It's just common sense.

To travel without a Schengen Visa and hope they won't stop you because you have residency in a given country is something an individual can chose to do if they wish. However as I read the OPs post, s/he wants to know what is legally allowed, since s/he mentions contacting an embassy for an official response.

Legally the OP needs a Schengen Visa.

I still haven't got round to getting an official answer, but in response to #7, the Schengen agreement allows for an individual to apply for a tourist visa that is granted by one of the Schengen states on behalf of the entire Schengen region. So, for example, if Italy sees fit to grant me a tourist visa, then France will accept Italy's judgement and let me pay them a visit too.

Using the same logic (I'm not saying that logic should apply here, but let's just pretend) if Denmark has decided that I am fit to live in their country, it would be strange if this has no bearing whatsoever on the other Schengen states. Also, I would be surprised if maaraw had so few hassles if s/he was travelling illegally.

Wayworn1, you are very adamant in your stance, but do you not agree that there is a little bit of a grey area here?

#8 - what you say is true for the Schengen tourist visa. However, student visas, working visas etc. are usually national visas, only valid for that particular country. What does your Danish visa say? Does it say valid for all Schengen countries or only for Denmark?
Still, I guess since you have a residence permit for Denmark, it should be easy enough for you to get a Schengen visa for travel since you don't really pose a "danger" as a possible illegal immigrant. Why don't you contact the embassy of the first Schengen country you intend to visit besides Denmark and ask them?