Was curious how the 'Special Forces' of the Spartans operated during their time. What was their main use? How did they train? How were the soldiers chosen?
Sparta had no Special Forces. In fact, to have Special Forces you need to have a regular, professional army, and Sparta didn't have that either.
The vocabulary of the modern military is mostly useless to describe Ancient Greek armies, because modern terms have no true parallels and usually come with a lot of anachronistic associations. For instance, the term "Special Forces" makes us think of covert operations, stealth tactics, high-tech equipment and the like. Sparta knew pretty much nothing of this. The Spartan army was nothing more than the adult male Spartan population conscripted in wartime; when there was no war, there was no army. Like all Classical Greek city-state militias, the Spartan army was disbanded after each campaign and its members returned to their normal civilian lives. Spartans thought of themselves as citizens first, warriors second; they would likely be gravely offended by the suggestion that they were soldiers (i.e. men who fought for pay, like mercenaries). It follows that there were no elite Spartan soldiers either.
A couple of Spartan units are usually associated in modern works with the notion of "Special Forces", though in all cases the term is wildly inappropriate:
The Spartan hippeis ("horsemen"), a royal guard of 300 hand-picked young Spartiates, who served as an honour guard in the service of the state and protected the kings in battle. They were selected from each class of Spartiates that reached adulthood, mainly on the grounds of physical fitness.
The Skiritai, a local levy used in particular combat roles. These troops are mentioned in a couple of sources, but unfortunately they all disagree about their specific roles. They seem to have been used initially as the guards of the left flank in battle; later, they increasingly took on a screening role for the marching column, suggesting they may have been light infantry. They were not Spartans, but Arkadians conscripted from a border region of Lakedaimon.
The kryptes ("hidden ones"), a small group of select adolescent Spartiates whose role is very poorly understood. Early sources say that a few Spartiates were chosen on turning 18, to spend their next few years in athletic training, ending with a brief "time in the wild" that was supposed to serve as their rite of passage into adulthood. This was known as the krypteia. Later sources say they spent this time in the wild hunting helots, as a way to surpress this slave underclass through terror. Even later sources suggest they operated as a mobile reserve in battle, but this is only noted once and we cannot see if it was the result of a change in the institution or if this was an exceptional case.