Sparta did indeed gain a reputation as the bane of tyrants at the very end of the Archaic period (the latter half of the 6th century BC). Thucydides is our earliest source to present them as such, and explains it as a result of their own internal stability, which was the main reason Sparta was admired in Antiquity:
But at last a time came when the tyrants of Athens and the far older tyrannies of the rest of Hellas were, with the exception of those in Sicily, once and for all put down by Lakedaimon; for this city (...) at a very early period obtained good laws, and enjoyed a freedom from tyrants which was unbroken; it has possessed the same form of government for more than four hundred years, reckoning to the end of the late [i.e. the Peloponnesian] war, and has thus been in a position to arrange the affairs of other states.
-- Thuc. 1.18.1
However, this sweeping statement may seem a bit unsatisfactory, since Thucydides doesn't bother to provide any examples besides Athens. Thankfully, Plutarch is always eager to supply information about Sparta, dubiously sourced though it may be. In his rhetorical treatise about the flaws of Herodotos, he accuses this early historian of failing to recognise the greatness of Sparta and its selfless crusade against tyranny, instead explaining its campaign against the tyrant Polykrates of Samos as the result of some trivial transgressions. For Plutarch, the actual pattern is clear:
For what breastplate or cup was the cause of their driving the Kypselidai out of Corinth and Ambrakia, Lygdamis out of Naxos, the children of Peisistratos out of Athens, Aischines out of Sikyon, Symmachos out of Thasos, Aulis out of Phokis, and Aristogenes out of Miletos; and of their overthrowing the dominant powers of Thessaly, pulling down Aristomedes and Angelos by the help of King Leotychidas? All of this has been described in more detail elsewhere.
-- Plutarch, On the Malice of Herodotos 21 (Moralia 859d)
Thank you Plutarch! That's a neat list of tyrannies overthrown by Sparta. The only problem is with that last claim - "all of this has been described in more detail elsewhere". This may have been true for Plutarch, who had a whole library of works at his fingertips that have since been lost. For us, most of the events he alludes to are not described anywhere, and we are left wondering how many of them are really true.
The case we are most certain about is the one mentioned by Thucydides: the overthrow of the tyrant Hippias, son of Peisistratos, at Athens in 510 BC. Herodotos and others give us a detailed account of the role of the Spartans in these events. But was Sparta really going around the Greek world ousting one tyrant after another, or did the Athenians merely grant them such an image in recognition of the help they received?
Herodotos gives an account of the Spartan campaign against Polykrates, which is the one Plutarch is criticising with his list. We know they did mount a major naval campaign to overthrow this tyrant and install a group of exiles as the new rulers of Samos. However, their assault and siege failed, and Polykrates held on to his power for 3 more years.
The other examples offered by Plutarch are barely known from any other material. We have a fragment of a lost text that claims Sparta overthrew the tyrants of Sikyon and surrounding states during the reign of Anaxandridas (the father of Leonidas and the first 'historical' Spartan king), c. 550-520 BC. Lygdamis was tyrant of Naxos around this time, so it's possible that the Spartans under Anaxandridas mounted an expedition to the island to oust him as well. A stray remark in Herodotos informs us that a Spartan army under king Kleomenes (older half-brother of Leonidas) was in Boiotia in 519 BC for unknown reasons, and this may have had something to do with overthrowing a tyranny in nearby Phokis, if we wish to be generous to Plutarch. Herodotos also tells us about the march of Leotychidas into Thessaly, which took place at some point in the 470s BC, but he claims Leotychidas took a bribe and aborted the campaign, so it's not likely that he "overthrew the dominant powers of Thessaly" as Plutarch claims. Meanwhile, the campaigns against Corinth, Thasos and Miletos are otherwise unknown. The first of these would have to have taken place generations before the others (the Kypselids were overthrown around 580 BC) and it's not very likely that Sparta would have ventured that far north so early.
In short, Sparta had a reputation for being involved in a number of campaigns against tyrants in the second half of the 6th and first half of the 5th century BC, though it seems not all of them really happened, and others were unsuccessful. For Athenian authors it was enough that Sparta definitely did have a hand in the overthrow of "their" tyrant Hippias.
But the Athenian case also demonstrates why the Spartans bothered to fight tyrants elsewhere. After Hippias was ousted, the Spartans under Kleomenes kept a tight rein on developments in Athens, actively supporting their champion Isagoras even to the point of backing his own bid for a renewed tyranny. The point, for them, was clearly not to eradicate tyranny as a form of government, but to use the overthrow of tyrannies as a justifiable and righteous pretext for installing governments more favourable to Sparta. There's no doubt that the Samian exiles that Sparta meant to restore to power on Samos would have been expected to become loyal allies, just like Athens - and that attempts to form untrustworthy new-fangled government systems like democracy would be swiftly crushed if Sparta could help it. In other words, their reputation as the bane of tyrants allowed Sparta to operate beyond their traditional sphere of influence and to install friendly regimes without being accused of subjecting Greek states to their growing hegemonic power.