Lacedaemonians/History

Sparta was a harsh military culture based on high military discipline. Parents were forced to let their boys leave the house at age seven and receive training at a strict military camp — if the boys were lucky enough to make it to the age of seven, that is. The Spartans had a cruel practice by which all baby boys were examined by community leaders immediately after birth. If the child appeared weak or deformed in any way, then he was taken into a remote area and left to die from exposure or wild animals. The Spartans did not want any member of their military to be anything less than 100% fit.
There were rigid social class structures in Sparta. The aristocrats, called the "spartiates", were people descended from the Dorian invaders of the second century. The second class of citizens, called "perioeci", consisted of landowners, artisans, and traders. They were primarily relatives of the natives who had lived in the region before the Dorians conquered it. The third class was comprised of conquered people who were basically serfs and were known as "helots", and lastly there were slaves in Sparta—usually prisoners of war. The last two classes did not enjoy the rights of citizens. At times it was even legal for citizens to kill the non-citizens (the helots) to reduce population!
The helots did the farming and were 4/5ths of the overall population. The Spartans were not the brightest people in the world: they forbade the use of gold and silver for money, and instead used heavy iron bars as money! For most Greek city-states, trade consisted of making olive oil and wine and exchanging it with surrounding regions to obtain timber and grain in return.
Spartans did serve as mercenaries, but they were one of the less common sights on Greek battlefields. They were sought after (particularly as commanders) but relatively scarce. Spartans had to receive permission from the ephors to serve abroad, which tended to keep a limit on the number of Spartan mercenaries; foreigners looking to recruit in Spartan territory also needed permission. Moreover Spartans as a group were less vulnerable to the pressures that inclined many other Greeks to serve as soldiers of fortune.
Examples
There are several known examples of Spartan mercenaries. The most famous is Agesilaus II, a Spartan king who led a mercenary army in Egypt — trying to earn money for a cash-strapped and weakened Sparta in the aftermath of the Spartan losses as Leuctra and Mantinea. The famous expedition of the Ten Thousand — mercenaries who supported Cyrus the Younger’s bid for the Persian throne — included about 700 Spartan hoplites and several Spartan officers, including Clearchus as the overall commander. However the muster roll of the Ten Thousand illustrates the relative scarcity of Spartan mercenaries: Spartans only formed less than 10% of the force, despite their prominence as commanders.
It’s also worth noting that both the mercenary stint of Agesilaus and that of Clearchus took place in the immediate aftermath of major wars (the Corinthian War and the Peloponessian war, respectively) when there were large numbers of displaced persons and demobilized soldiers all over the Greek world. In more settled times it’s less clear why an ordinary Spartan would pursue a mercenary career.
Social background
Many of the other parts of the Greek world produced mercenaries out of desperation: if you were a younger son, unlikely to inherit enough to live on, setting out for adventure made a rough kind of sense. This was particularly true because the Greek world did not offer many paid occupations for someone who aspired to social status: owning land and plying a spear were really the only respectable jobs for gentlemen. Working with your hands or practicing a craft was for members of the lower orders. So, many poorer members of the gentry (particularly younger sons, who might not inherit enough to maintain themselves) took up arms as a last resort.
In Sparta, however, there was no need to worry about inheritance. Any Spartan male who was accepted into the citizen body was assigned a state-owned farmstead (a “kleros”) and helot workers to maintain it. Instead of inheriting from his father every Spartan got his position in life from the state. While it’s true that the Spartiate lived a life that other Greeks regarded as, well, spartan he was nonetheless guaranteed an income, a social position and freedom from worries about poverty in a way that most Greeks would envy. Thus ordinary Spartans had less incentive to go roving than, say, their neighbors in Arcadia whose meager countryside produced a lot of mercenaries. The primary exception were Spartan exiles: men like Clearchus who had fallen afoul of domestic politics and could not remain at home in Sparta.
Spartan mercenary generals
One pattern that does emerge is the popularity of Spartan commanders. As more-or-less professional soldiers in a world of amateur citizen militias, Spartans were sought after as commanders. The “Spartan” army always included significant contingents of allies, so many Spartans had experience leading foreign troops. Combined with Sparta’s military mystique, this made Spartan commanders particularly attractive to those seeking to employ mercenaries, even when the troops themselves were more heterogeneous. Examples of Spartan mercenary generals include Xanthippus, who led Greek mercenaries in the service of Carthage, Cleonymus who organized the defense of Tarentum against the Lucanians, and the exiled Spartan general Cleandridas who also operated in southern Italy.
Ironically, although Spartans didn’t dominate the mercenary armies of the classical period the most active recruiting ground was at Taenarum in southern Laconia. As a place where the north-south sea route to Egypt and the east-west route from Greek Sicily to Asia Minor crossed Taenarum became a notable mercenary shopping center: in 303 Cleonymus raised 5,000 troops there for his Italian expedition. From the perspective of the Spartan state it had an additional advantage: despite being Spartan territory, overland communications with the rest of Laconia were difficult, which kept the mercenaries effectively quarantined.