Wsrtjtyk

The primary advantage would be surprise.

I'm trained in eastern martial arts including swordfighting. I was trained to not look at the weapon, but at the body of the opponent. The weapon moves too fast and will confuse your focus, but upper body and arm movements telegraph the sword strike a long time before it actually happens. A well-trained opponent may not even notice that your sword is invisible since he isn't even looking at it.(*)

But at the beginning of the battle, the opponent would not realize that you are holding a weapon at all. You might want to wrap something like a paper scroll or something around the handle to mask your unnatural hand grip, but especially if you gesticulate a lot while speaking (for safety reasons, primarily with your other hand), you could well land a nice swing before your enemy even understands that you are holding a weapon at all.

In a prolonged fight or battle, there would be a small advantage, but the ideas outlined in other answers to hold a smaller weapon in addition so that enemies use a false distance assumption is probably the best use.

(*) anyone who answered that it would make parrying difficult if you can't see the blade doesn't have swordfighting training. I don't need to see your blade to parry it, I only need to know the direction of the swing. I would not try to parry a thrust, ever, I would sidestep it. Again I don't need to see the blade, the arm tells me enough about where the thrust goes. This might be different in western-style swordfighting, but I can't imagine the difference is huge.

Bald Bear basically nailed everything I was going to answer except for one particular other use I saw once in an anime called Get Backers.

In the episode I watched, a bad guy got into a swordfight with one of the main characters. The hero kept being hurt by the sword even though he as pretty confident in his skill to dodge the blows. Still, he kept being hit time after time.

The explanation for that is that the attacker's sword was actually a few inches longer than it appeared. The additional inches were invisible, which gave him an incredible advantage early on - if one of these attacks got, say, his throat, that would have been the end of the fight.

A very good fighter is able to gauge their opponent's range and uses that to both predict/defend against attacks as well as strategizing their own attacks. An invisible sword would be much harder to gauge and thus create a significant advantage for the wielder. This was used in the anime series "Fate/stay Night" where one of Saber's special power was being able to shield the length of her sword.

The primary advantage would be in 1 on 1 duel type settings. This would have little advantage in a large group skirmish as one rarely has time to stop and closely evaluate range in that kind of setting. Additionally, it would actually be a determent in an intimidation type scenario. With intimidation, size matters. Since you can't show just how impressive your weapon is, you can't use it to intimidate.

To add onto the other answers, if its a one handed sword you could easily fake out your next attacks in duels by grabbing the sword with both hands and then pretending to hold it in both , this would make it incredibly difficult for an opponent to block or parry attacks. For example you could raise one hand and pretend to do a slashing strike with your empty hand causing your opponent to try and block it, and then use the actual sword hand to for the real attack.

Naively thinking, it would be the ultimate super weapon since you can hardly defend against a weapon that you can't see, but realistically it's far from that.

It may be great for murdering someone and for carrying it to places where weapons are banned. But in a fight, it would not be much of an advantage.

Many medieval sword techniques were surprisingly sophisticated and do not actually require that you see the opponent's sword (much like e.g. you do not need to see the opponent's hands in some martial arts, which once upon a time I thought was something very special and awesome, but it's actually no big deal!).

An experienced fighter (you could kill an unexperienced one with a visible sword anyway!) will know what move to make based on your stance and arm movements, and it will work as good as if he could see the blade. You don't swing and bang blades together repeatedly like in the movies. You make a move which guarantees contact, and after that everything is about feeling and steering, while keeping contact. No sight necessary.

On the other hand, if the blade is invisible, you cannot see it either, which isn't necessarily an advantage. Try and sheath it without cutting off a finger.

Plus, blood (or dirt) that may stick to the blade will be visible, even if the blade itself is not, so the trick's efficiency is limited.

The late medieval period was characterized by the movement away from full plate armour and combat swinging slashing/chopping swords and crushing/penetrating maces and hammers, toward unarmoured combat, armed with thrusting swords such as rapiers and foils, and light cut/thrust weapons such as sabres.

As Tom has pointed out, when using a slashing sword, whether it is a broadsword or a katana, a long stroke is required to cut through armour, and even if the opponent is 'unarmoured', unless the blade is razor sharp, even ordinary clothing - especially if made from silk, as would be the case for samurai - could prevent serious injury.

In such a case, as Tom has pointed out, even to an indifferently trained swordsman, even if an opponent's blade was invisible, that opponent's movements would be declaring their intentions quite well enough to defend against them to a reasonable degree.

However...!

The European movement toward unarmoured combat - that came to pass due to the rise of firearms in battle that made all but the heaviest, most expensive armour effectively useless - led to the invention of thrusting swords.

These weapons were designed for sheer speed - life or death was measured in fractions of a second, inches of movement and surprisingly little force. Three inches of extension and 100g of pressure from the point of a thrusting sword against an unarmoured opponent was the difference between life and death, when delivered to a vital area.

When fencing with western thrusting swords, the difference between a successful parry and an unsuccessful one could be a matter of a few inches of movement of the tip of the sword, probably no more than 4", possibly less.

As someone with training in modern fencing, I can say that being able to see the position of the tip of the opponent's sword is of vital importance - my instructors made that quite clear. Since the sword's tip need only move 3 or 4 inches in order to avoid an opponent's parry, and the sword may be a metre long, very little movement of the hand or arm is needed in order to achieve that movement. Since one need only apply around 100g of pressure in order to penetrate clothing and flesh with a needle-sharp blade, and need only penetrate the opponent's body to a depth of 3-4" in order to cause a potentially fatal injury, the only obvious part of an attack is the thrust itself, since there is no magnification of movement caused by angles over distance.

Since a fencing blade need only be held very lightly, a moderately skilled fencer can conceal the slight movement of his arm and hand with even slighter movement of the fingers. This is what makes being able to see the position of the tip of the opponent's sword so vital.

An invisible sword would be of no use in a formal duel. No honorable second would allow the use of such a weapon (as they take charge of the weapons until it is time for the duellers to take them up), and somehow managing to use it would instantly brand the wielder as a person completely without honor, and likely lead to their being quietly stabbed in the back in some secluded location at a later date.

However, in an undisciplined street brawl or upon a battlefield, while an invisible sabre or katana would be merely somewhat disconcerting to the opponents unless they were untrained, an invisible rapier would be utterly terrifying even to the most highly trained opponents. The wielder would, with only a little practice, be easily able to handle the sword's invisible nature and parry an attack, but the opponents would be guessing blindly, quite literally, when it came their turn to parry, and with only a little skill, a simple disengage against a lucky parry could reverse the situation. Even master fencers could fall quickly to an indifferently trained opponent wielding such a weapon. An invisible rapier might allow an indifferent fencer to prevail against odds of 2:1 in all but the most unfavorable situations, and in favorable circumstances, they might stack up the corpses of their opponents in piles too great for their comrades to climb across. Were that to happen, no doubt archers, arbalestiers, musketeers or even slingers would be called so tha the swordsman with the invisible blade could be shot down from a safe distance - if any were available - or an armoured swordsman could be called, against whom an invisible blade would be at even more of a disadvantage since the wielder of the invisible blade would not be able to aim for weak points in their opponent's armour as well as if their blade was visible. Otherwise, one man with an invisible blade and a narrow passage to defend could cause an entire army to retreat, as long as their endurance held out.

Fencing is surprisingly exhausting. The constant movement is a good cardio workout, so as long as opponents were willing to keep coming, even after the comrades who preceded them had all fallen, sooner or later, the defender will become exhausted and will be unable to continue defending. If they have any sense, they'll retreat before that point, and if defence of their position is a matter of life and death, having 2 to 5 comrades, all of whom have practised with the invisible blade, could allow the defenders to hold off an army of unarmoured swordsmen indefinitely. They just need to hope that their assailants don't think of throwing rocks...

One other advantage of this sword is that your opponent doesn't see your grip on the sword, meaning that with unconventional grips, your opponent may not be able to tell where the blade is coming at him. Even better, you can use feints that wouldn't otherwise be possible, because your opponent has to attempt to block every stroke you use, even when the blade may be nowhere near him, causig him to expect a block, then be put off balance when he meets no resistance.

You ask for situations of a duel or a skirmish.

Useful additions-
A sword with a notch which can be used as a 'sword-breaker' could be useful to quickly disarm the opponent in a swordfight, especially if the adventurer is careful to keep the presence of the notch secret until his chance comes. SO he basically will have a one edged sword, but he only has to turn it around and the opponent's sword will get caught in a notch. TBH I do not know how exactly sword breakers work, but I think most opponents will not consider the possibility that their sword will get stuck in another sword, even if they consider the same about the protective gear.
Note that sword breakers are usually shorter and thicker than other swords, but an invisible notch would definitely be a surprise.
! [sword breaker] (https://www.medievalcollectibles.com/images/Product/medium/ED2206.png) ! [

Additional Benefit
One aspect that other answers do not cover is- An invisible sword can be brought into places where swords are not allowed or where people are frisked before entering.

NOTE: My answer assumes the adventurer can either see or sense the sword when it is away from them upto some range. When in hand, the adventurer can be used to the sword length by practicing. Possible solution- Using the magic you mentioned, there can be a bond between the adventurer and the sword. This will also prevent losing the sword.

NOTE: The adventurer may actually be better fitted with a visible sword and an invisible weapon. This concept is mentioned in a previous answer, but my suggestion is the invisible weapon be a dagger as wielding 2 swords may not be much easy in practice or may be easily noticed, a dagger being used is not easy to notice(most assassins I have seen or read used daggers instead of swords).

You can throw it.

If it is a sword, rather than a dagger, then you can reforge it into a dagger.

He can practice throwing by simply covering the dagger with mud.

Once he can hit targets, he can kill anybody from middle range because the enemy won't try to dodge the dagger since he can't see it.

Additionally, you can split the sword into thin pieces instead of a single dagger to throw during bandit raids.

The invisible sword has a major disadvantage: No competent friend would dare help its owner in a melee. This means that its owner is more likely to be outnumbered, and will never be able to outnumber an opponent in close-in fighting.