When do we need to put a comma after "so" at the beginning of a sentence?

I noticed that most of the times when the conjunction "so" is used at the beginning of a sentence, it is followed by a comma:

So, this gets published but the fact that it is inaccurate gets moderated out.

Occasionally, I find sentences with no comma after "so":

So he went to Ishmael and married Mahalath.

I am guessing that whether to put comma or not after the "so" conjunction is related to the context and emphasis rather than to the sentence structure, but I am unable to pinpoint it out.

So when do we need to put comma after "so"? (← Do I need a comma after the "so" here?)

P.S.: I know there is another usage of "so" as an adjective or an adverb, in which case no comma is necessary, for example: "So big is the caravan that it cannot fit into the garage."

P.S.: I did skim through the list of questions with [comma] tag, and found no question on this conjunction yet.

So, now that you've asked this question, how can we answer it? "So," suggests a substantial turning point in the discourse, for example between describing a situation and reacting to change it. See Fattie's answer for good examples where the turning point is substantial because of the outrageousness of the situation. Less outrageous examples still have a substantial turning point:

As my students, you have worked hard and studied carefully. So, today it's time to party!

So now let me describe "So" without a comma. "So" suggests logical continuity, for example between describing a situation and its usual result. When possible, it would often be better to combine a "So" sentence with the preceding sentence.

As my students, you have worked hard and studied carefully. So I know you will pass the examination.

As my students, you have worked hard and studied carefully, so I know you will pass the examination.

If you are emulating a style of casual speech, as the others have explained, you could use a comma after so, especially if you hear a significant pause. Such a pause may be caused by indignation, hesitation, etc.

So, what do want to play with next?

Traditional style, on the other hand, forbids it. Whether or not a pause is heard doesn't matter. There is no one-on-one relation between pauses and commas: sometimes a pause is indicated by a comma, sometimes it isn't. Syntax plays a major part too.

Roman emperors were often adopted by their predecessors. So Augustus adopted Tiberius, and Trajan is said to have named Hadrian his successor shortly before his death.

Note that the use of so to denote consequence without either and ("and so...") or that ("so that...") is still less formal than and so or so that. In the above example, it denotes similarity, not consequence.

If so is followed by a parenthetical phrase, a comma naturally follows in either style (though some authors find this comma burdensome and leave it out).

And so, ignoring the senate's command, Caesar crossed the Rubicon.