Do we know if it discovered/developed in a single location and spread or was it a natural progression to the stone tool that was developed in many locations independently?
"Every tool is a weapon, if you hold it right."
Early weapons were whatever dangerous objects the fighter had handy from their regular work day. Hunters had spears and arrows. Woodsmen had axes. Farmers had sickles for harvesting. Hunters and farmers both had knives for butchering. All of these tools have antecedents going back into the stone age.
Swords evolved from these tools, but would not have been thought of as swords (as a distinct sort of object, in other words) until that evolution had transformed the tool sufficiently that it was now considered a weapon foremost. Generally this involved an increase in length such that the tool was now more suited to swinging at an enemy who stood out of reach, rather than cutting something that was held in your hand.
This evolution happened early enough that it verges on prehistoric--before there were written records explaining what the inventors of swords were thinking--so this is more of an archaeological question than a historical one. A few cultures have managed to pull off swords using stone or neolithic technologies. The most well-known examples are the Aztec obsidian sword-clubs (macuahuitl) and the Polynesian shark-tooth clubs (leiomano). But both of those examples actually date later than the earliest metal swords known, so they don't really help us to understand who came up with the idea first, except to raise the possibility that it might go back very far indeed. But if we ignore short stone daggers, the archaeology of swords finds its first examples in and around the Fertile Crescent.
Straight copper swords dated to roughly 3000 BCE have been unearthed in Arslantepe (Turkey), which resemble long daggers in their basic form. Copper is a relatively soft metal, so the practicality of these swords for combat is questionable. Bronze is a far better metal for weapons, and straight bronze swords became more common in southeast Europe around the 1800-1700 BCE. There is a large gap between those dates, though, so it's hard to say if or how these weapons were related, other than to note they both follow the general pattern of straight dagger-like blades.
Sickle-swords arose in ancient grain-farming cultures, and were being used militarily in ancient Egypt by the 18th Century BCE. Some, like the kopesh, were very sickle-like with extremely curved blades, but the resemblance to a sickle may just be coincidental. Others had comparatively straight and flat chopping blades at the end of a long haft. And even in the curved case, the edge was on the outside edge of the curve, opposite to an actual sickle. These weapons were essentially battle axes evolved into a more sword-like format.
That suggests that sickle-sword is a misnomer, and the association with grain-farming regions was coincidental, but true sickle-swords did exist. The most famous examples are probably the falx and sica, curved swords/knives used by the Dacians of south-east Europe against the Romans. Like true sickles, they were sharpened on the inside edge of the blade. The Dacian examples are too late to be considered for earliest swords, but they do at least show that swords were evolving independently from nearly any hand-held tool with a sharp point or edge.
Incidentally, the word sica comes down to us today as "sickle", and the word falx comes down to us as "falchion", a sword that in its later forms is somewhat axe-like, with a broad chopping head sharpened on the outside edge. So you can see that the evolutionary lines are somewhat tangled up. The question of which was first gets a bit blurry when you understand that modern swords are actually a convergence of several different types of weapon.
Thank you for that answer.