So, as a preface, I'll (predictably) be using quite a few obscenities in the text below. At my own discretion, I'll be censoring outright slurs, but not general profanity.

I'll also use modern-day examples for the sake of clarity/comparison.

Some linguistic background
Supreme Court justice Potter Stewart is known for his quote "I know it when I see it, referring to how to define hardcore pornography, but as an aphorism it's been extended to talk about obscenity/profanity in general.

And in a sense it is true that the lines between outright profanity, vulgar and informal language and inappropriate speech acts are fuzzy, and depends on context and culture.

Linguistically, we talk about registers when we distinguish for example, formal vs informal vs vulgar or obscene language. In English, register is largely limited to vocabulary (copulate vs have sex vs screw vs fuck), but other languages (e.g. Japanese or Korean) encode politeness grammatically; register also includes acceptable speech acts (e.g. acceptability of interruption), non-verbal communication, etc.

On the level of obscenity, Steven Pinker in The Stuff of Thought (2007) distinguishes five functions of swearing: abusive (fuck you, asshole!) vs cathartic (ow fuck!) vs dysphemistic (look at that fucking asshole) vs emphatic (that was fucking awesome) vs idiomatic (fuck me, babe), with the latter being a sort of catch-all category for acceptable or even expected swearing in informal or intimate contexts.

If we combine these two concepts, we can see, of course, that fuzzy overlap: "you idiot" is an insult (abusive and informal), but not exactly swearing like "you dumb fuck" or the r-slur (abusive and obscene).

Taboos and taboo language
Now, where do we get our swear words and insults from? Cross-linguistically, it's drawn from semantic fields related to subjects that are considered taboo*. What is considered taboo, in turn, is culturally determined.

Common categories are related to sex (fuck; various terms related to promiscuity and sex work; cock, cunt), bodily functions (piss, shit; anatomically ass(hole)), comparisons to animals (b*tch (which partially overlaps with the sexual category)), mental or physical faculties (various ableist slurs), race and ethnicity (obviously the N-word, but of course there's a vast vocabulary of other racist slurs and ethnopaulisms), death and disease (not very productive in English, but compare fuck off and die, kill yourself; not swearing per se but interwoven with the topic), cleanliness (originally and still regionally slut, dirty), and religion (Christ!, goddamn, bloody, archaically gadzooks), and combinations thereof.

Different languages and cultures weigh those categories differently, however; German generally favors scatological swearing but also uses animal terminology much more systematically, Bavarian religious and animal vocabulary, Dutch delves into disease, Quebec French uses religious swearing intensely versus the more sexual Metropolitan French.

* As a sidenote, taboos and the links between the shameful, the forbidden and the sacred, are of course a whole nother topic of history/anthropology/sociology/sociolinguistics.

Sources and Latin as an example
Our best source for historical swearing, insults and taboo language are, of course, direct attestations. Furthermore we have references to tradition in later works, as well as etymological analysis.

There are, however, a few problems with that:
- Bias of surviving material; here we are limited to cultures that had writing, what they wrote and what survived, and as a rule of thumb, the further back we go, the less material we have
- lack of contemporary meta commentary about the words we know existed, so sometimes, we must at least partially conjecture how obscene a word would have been
- replacement of taboo words and shifts in acceptability make it hard to pinpoint when a word was how obscene.

To finally turn to concrete historical examples, let's look at Latin, for which we have a rich transmitted vocabulary.

The sheer number of words we have for genitalia alone is by itself taken as evidence that some of them were more obscene than others and not just poetic license. This is confirmed by a surviving letter by Cicero in which he discusses a number of obscene words without directly mentioning them. He doesn't necessarily judge the usage by his partner in correspondance, but prefers not to use them himself.
Notably, he quotes a Senate speech that accidentally sounded obscene ( "hanc culpam maiorem an illam dīcam?" - the last two words sounding like landica, the word for clitoris, one of the least commonly attested words in Latin; neither Catullus nor Martial use it despite their otherwise very explicit writings; it's mostly attested in graffiti and one medical text from the 6th century).
Cicero further alludes but does not use mentula, which seems to be the base word for penis/cock, but does use penis while mentioning its somewhat vulgar status (and that it is a metaphorical synonym, originally meaning tail).

Martial, for example, warns his readers in 11.16 "if you're too serious, reader, you better leave now (...) o how often will you strike your hard vein against the cloak (...) and you as well will read the wicked and playful words of our book while wet, girl, even if you may come from Padua" - while the warning can and probably should be read as satirical, we can infer the idea that the following content could be seen as filthy, and filth meant to arouse.

Catullus, meanwhile, infamously threatens two critics in c16 that he'll "fuck them in the ass and mouth", using the verbs paedico and irrumo, and uses two nouns pathicus and cinaedus, loans from Greek that both denote the passive partner in a male-male sexual encounter. So in modern translation, homophobic slurs.
Context tells us that these were certainly insults, but it is difficult to gauge how much offense they would have caused compared to modern (semi-)equivalents.

For a somewhat better look at actual spoken Latin, we can finally turn to graffiti. On the walls of brothels, we find numerous versions of "X hic futuit", "X fucked here".
One interesting attestation from Pompeii, and a more definite parallel to modern day swearing, would be CIL IV.1700 "ut merdas edatis qui scripseras sopionis" - "may you who draws [hard] cocks eat shit". (Martial 3.17 also offers us a metaphorical use of merda in reference to food: "sed nemo potuit tangere: merda fuit." : no one could touch it, it was shit).

Conclusion
My focus on Latin is due to me being most familiar with it and having more sources at hand/available; that is not to imply that we have no sources for older profanity. I know that there's at the very least ample material for Ancient Greek, and I'd certainly be interested to hear from someone knowledgeable about other ancient cultures.
But, bottom line, due to the nature of language attestation, we simply cannot tell when societies first developed swearing (in the sense of using obscenities).

Sources:
Cicero, Ad. Fam. 9.22 (translated version)
Adams, James N. "The Latin Sexual Vocabulary", 1982
Franklin, James L. “Games and a Lupanar: Prosopography of a Neighborhood in Ancient Pompeii.” The Classical Journal, vol. 81, no. 4, 1986
Holzbeck, Niklas "Catull. Der Dichter und sein erotisches Werk" C.H. Beck, München 2002