I have answered similar questions before. In this answer I discussed the use of swords in battle.

It is hard to say when in all of history the 'high point' for both the use and 'mastery' of swords. Questions like this that span the entire globe and all of history are difficult to answer both because their terms are unclear (what do we mean by 'mastery' of swords?) and because no one person has enough knowledge to compare disparate times and places.

So with that said, I am going to focus on late medieval and renaissance Europe and how swords were used then and there. In general, swords were commonly used in late medieval warfare and in civilian combats. This is because they were supremely flexible weapons that could be used in a wide variety of circumstances.

First, some background on the swords themselves. The later Middle Ages saw the price of swords plummet. While one can overstate the rarity of swords prior to this period, the increased availability of swords as the high middle ages progress into the Late Middle Ages is marked. This is part of a general drop in the price of steel, which is also seen in the increased availability and affordability of armour. Advances in metallurgy also mean that swords are more commonly made entirely of steel, and often of a single piece (rather than being welded together from several bars) This all-star construction allows for longer, pointerier, thinner swords. So swords are both much better (metallurgically) and more common in the later Middle Ages than they were earlier. This means that all elements of society (including tradesman and wealthier peasants) take up the use of swords both as sidearms in battle and as weapons in civilian life - for protection from bandits, to settle private quarrels, and as a kind of sport.

To add on to u/WARitter's excellent answer: it's very difficult to say for certain when or if there was ever a "high point" in the use of swords, but there is a confluence of social and technological changes that occurred around the mid-16th century that led to, in my opinion, a proliferation of fencing knowledge and activity.

The mid 16th century saw a number of newly printed fechtbucher and fencing treatises from other countries. The tradition, as u/WARitter points out, was not new. Treatises had been in circulation for a long time, and many of the German manuals followed from a zettel (usually translated as "recitation) of a man named Johannes Liechtenauer, who had apparently journeyed throughout Europe studying fencing in all its forms, and put his knowledge down in that form.

A large proportion of the earlier fechtbucher are directly related to Liechtenauer's zettel, providing glosses of certain lines, and adding images to the text.

By the mid 16th century, with the proliferation of printing presses and the apparent popularity of fencing manuals, many more started appearing. I'm going to focus on just a couple here that are relevant to your question: Joachim Meyer and Paulus Hector Mair in Germany before shifting gears and talking about Iberia and Italy.

Both Meyer and Mair are pretty interesting men in their own right. Meyer was a citizen of Strassburg (through marriage to a widow, interestingly enough) which is important to his activities as a fencer: citizenship bestowed privileges (like the ability to carry weapons), but also had expectations of duty and service to the community. Many of those expectations are wrapped up in the idea of "gentleman" rights and duties, which are far too complicated to really explore here, but suffice to say that there were many, many more armed men walking around places like Strassburg than there may have been a hundred or two hundred years before. Wearing a weapon was a privilege that many took very seriously.

In any case, Meyer's status as a citizen allowed him to be armed, and we know that he worked as a cutler, and was a member of The Company of Smiths, a kind of general guild for metalworkers. Although we can only guess as to his training and education, we know that Meyer was an accomplished martial artist, and requested permission several times throughout the 1560s for permission to hold fencing demonstrations and competitions called "fechtschulen."

Fechtschulen were a non-lethal demonstration of martial skill and prowess, and would have featured wrestling in addition to fencing with a variety of swords: Meyer's 1560 and 1570 fencing treatises featured the longsword (a "bastard sword" is probably the most basic pop-culture relative), the dusack (think cutlass or pirate sword; a relatively short single-edged curved sword with a complex hilt), and the rappier or sidesword. Polearms and dagger techniques may also have been demonstrated. There may have even been shooting competitions.

Meyer himself writes at length about the necessity of a gentleman to be well-versed with all of these weapons, and lays out his treatise in a manner that reflected that: the longsword was the basis of all combat, the dusack was the basis for all weapons in one hand, and the rappier was a necessity for a German gentlemen due to their popularity elsewhere - a man must know how to defend himself with it.

But his manual also addresses a number of interesting social issues: thrusting, for instance, was "abolished amongst us Germans" and was not taught in the longsword section, but was taught in every other section. The reasons for this stem from both the fechschule culture which saw men fighting "to the bloom" or to first blood with foiled longswords (known as feders now, though there's some quibbling about the specific term), and with the idea of legal protection. A thrust even with a feder could easily be lethal in practice or sport, and, in a deadly situation like a duel, a thrust may have been legal basis for murder rather than self-defense.

In any case, Meyer's treastise and his fencing experience tells us that the early modern German gentleman had to be well-versed in a variety of weapons - at least three swords, in addition to dagger, wrestling, polearms, and firearms - and that there was a social structure in place that encouraged the practice of them.

Mair, similarly, included a huge variety of weapons in his own 1542 treatise: longsword, dusack, short staff, long staff, halberd, scythe, flail, peasant staff, sickle, and mixed weapons. A second volume covers grappling, dagger, sidesword, sword and buckler, polexe, armored fencing, "long shield," and mounted combat.

Obviously, one was not expected to be a master of all of these, and Mair himself appears to be martially competent, but more importantly, he was a collector of older fechtbucher and somewhat of a historian interested in fencing primarily. Many of the surviving fechtbucher we have available today are actually books owned by Mair and annotated by him, before he included many of their teachings in his own compendium.

He also can't resist informing his readers a bit on the purpose and history of fencing:

And while many of the learned say that this art of the knightly sport, as other arts beside, must have come among men to influence[?] their appetites and pleasure, from above, i.e. from God and celestial influence of the stars, as is well believable. Besides this, some say that Pollux, who was honoured by the Romans, was an instigator of this honourable art; others would attribute the honour of such invention to Mercury. But both these statements must be found somewhat obscure and uninstructive from the fact that they do not explain what use or profit they would have made from this art, or which lords they took as their disciples that would have learned the art from them and in turn passed it on.

But the majority of the same historiographers state and testify that Probas, the famous fencer and teacher of Theseus, the king of Athens in Greece, in which realm the knightly art in the beginning and for a long time thereafter did much prosper, was the first inventor and establisher of this art.

Any citizen in early modern Germany was expected to serve in the militia, and was expected to back up their claims to social class with violence. There was a flourishing sport culture around fencing similar to our modern conception of the idea, and duels, disputes, and small-scale warfare were present threats to a gentleman's safety. It stands to reason that many gentleman would have been very well schooled by masters like Meyer or Mair to prepare for those possibilities.

By the early 17th century, similar things were happening in Italy. We have a huge number of surviving fencing treatises from Italy and the Iberian peninsula from masters such as Nicoletto Giganti, Salvator Fabris, Ridolfo Capo Ferro, and Gerard Thibault. In the Iberian peninsula, the system of fencing later known as "destreza" was forming around the same time that Meyer was teaching his fechtschuler.

There were similar social systems in place both in Italy and Iberia, that expected gentlemen to serve as social guardians of their class, and to some extent in an organized military capacity.

I feel like if I keep going I'll never stop, but short answer: look to the mid 16th century for evidence of sword culture, and you'll find it just about everywhere in Europe.

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Firstly, check out the excellent website Wiktenauer, which is a wikipedia of sorts of fencing manuals and historical articles. It was built for the Historical European Martial Arts community, but is quite historically rigorous, and many of its sources are the primary source material.

Otherwise, two modern publications talk more about Joachim Meyer, both by Jeffrey Forgeng: The Art of Combat which is Meyer's much more complete 1570 treatise, and The Art of Sword Combat which is Meyer's 1560 treatise.

I do not believe that a publication of Mair's work exists, but the wiktenauer page is very detailed: Paulus Hector Mair

For more on the social customs of armed violence in Germany, B. Ann Tlusty's excellent The Martial Ethic in Early Modern Germany is irreplaceable.