Mutually intelligible is not an exact thing. From my own experience:
A native speaker of Danish, I was able to speak with my Norwegian housemate. It worked OK, but not great, and not for anything technical (like science and finance, where the Germanic words are very different to the English).
With my Swedish friends however, it was harder. Not sure why, since Sweden is visible from Denmark but Norway is a bit further away. Also, I found the TV Series "The Bridge / Broen" quite unlikely. Swedes and Danes don't talk that easily together, and certainly not with slang. The scenario of the cops from opposite sides of the bridge working together could still work, but mostly in writing. Also, the cadence is very different. Swedish is melodic. Danish is grunty.
Of course I have had no instruction in Swedish or Norwegian, but it wouldn't be hard, based on friends who have moved.
German is not mutually intelligible with Danish. But it is. I went to a handful of German classes, and it dawned on me that changing a few things around makes High German pretty much like old Danish, like what you'd hear in old black-and-white movies from around the war. These days Danish has a lot of English in it, and something about the way sentences are put together makes old movies seem very... old. But once you have a simple map in your mind, Danish appears to be German with some substitutions and simplified grammar. I can read the paper in German now, despite having had just those few lessons.
Swiss German, now there's a weird one. The sounds are different. I suppose I'm twice removed from it, but even native Germans I know will produce the WTF face when they hear a Swiss German. Words and grammar are similar, but different enough to be recognisably so by a foreigner like myself. But I suppose it's like Swedish and Danish.
Apropos the article, Singlish I found quite interesting. To me it's just English with sprinkled Chinese, and I think most English speaking people will not have a problem understanding it without instruction. One useful addition: English doesn't make much use of the modal particle, so it makes sense that Cantonese "la" is heard all over the place.