My answer to this question is going to be relevant to late Medieval western Europe, and probably won't be relevant to other places and times, but I hope its a good start. My answer is based on Michael Chidester's article on dueling in the 15th century, itself part of the preamble to his excellent translation of the Fior di Battaglia MS M 383.

So a late Medieval duel is identified as becoming a separate phenomenon from trial by combat in the 13th century as a way of settling disputes while on a military campaign: an Army would construct an enclosure within which two combatants would settle their differences. This became folded into folded into civil law as a way for grievances to be settled outside a campaign: The duel was conceived of as a battle between the two parties, occurring (mostly, we'll get to this later) within the law. I'll use an example of a duel which demonstrates a lot of the legal and martial issues which had to be accounted for in order to get a duel 'off the ground' to show just how structured and regulated the practice could be. This is taken from an account of the duel by one of the participants.

An Italian Knight named Galeazzo Cattaneo Dei Grumelli (sometimes Galeazzo Da Mantua) found himself visiting the court of the king of France where, in discussion with other members of the great and the good attending court, a famous French knight named Jean II Le Maingre (often known by his nickname Boucicaut) said some insulting words about Italians. Taking offence, Sir Galeazzo got into a heated verbal exchange with the Frenchman. Their exchange ends with this description "Ultimately nothing could be done because, each one wanting to keep his honour and show what he had said was true, Sir Galeazzo asked for battle."

So here we see that this isn't a Trial by combat in the sense of it being a way to settle a legal issue such as a property dispute or to try a murder accused and accuser. Rather, Galeazzo was offended by Boucicaut, and after an attempt by other members of the French court to get them to settle their disagreement, Galeazzo challenged Boucicaut to fight him.

What happens next is a classic part of a Medieval duel, and probably seems odd to us, given we are more familiar with the more modern honour duel which takes place outside of legal sanction. The king of France was asked by Boucicaut if he would provide legal sanction for the duel and a place for it to take place... and the King of France refused. So Galeazzo left France, went to the town of Mantua and petitioned the Lord there for sanction and a place to conduct the fight... and he refused as well! So Galeazzo finally went to Padua and there he found a lord willing to grant the duel legal standing. So Galeazzo wrote a letter to Boucicaut and told him to show up in Padua on 15th of of August if he wanted to maintain the truth of what he had said.

The duel was basically a sporting event, the Lord of Padua set up a wooden tower, wooden bleachers, wooden terraces constructed on the roofs of house which overlooked the square, and these, we are told were rented out at a fee of 3 Ducats for a seat.

When the fighters entered the field there seems to have been a misunderstanding about the proper procedure, Boucicaut approached Galeazzo on his horse, Galeazzo, however, stayed on the ground beside his horse, expecting Boucicaut to throw down his glove in challenge. Potentially confused, Boucicaut dismounted, had his spurs removed, and, thinking the glove had been thrown down, approached Galeazzo on foot, armed with a spear. After a short exchange of blows, the lords of Padua and Mantua called off the match, and managed to bring Galeazzo and Boucicaut to an agreement, after which they kissed and embraced each other.

This is a good template for a legal duel: an insult is percieved and met with a challenge - the challenged party (here Boucicaut), is given the opportunity to find a place for the duel, which doesn't work out, so the challenger is given opportunity to find a place to hold the duel. The duel is a public proceeding, and apparently in this case a very popular one. However, the Lord who gives the duel license is the ultimate authority, and, as we can see here, calls off the battle before any real damage can be done, and between the Lords of Padua and Mantua the egos of the two fighters are assuaged and they kiss and make up.

This is how a Medieval duel plays out ideally: A battle was granted the legal go-ahead by an overlord and took place in armour, often for a set number of strikes with specific weapons. An example procedure would be 3 passes on horseback with a lance, three passes on foot with a weapon such as the pollaxe or the sword, and then three passes with the dagger.

One could however, step outside the law. An Italian legal scholar called Paride del Pozzo in a 1472 legal treatise on dueling goes over the 'case law' for lack of a better word about duels which occur outside of legal sanction. The thing is, if it's not legal, why does he bother discussing it? It seems that plenty of men wanted to fight duels without legal sanction, resorting to going into secluded woods and places where they wouldn't be observed breaking the law. Pozzo makes the case that an honourable knight, if challenged to a duel and legal sanction cannot be found, does not need to go into the woods and fight a hidden duel, for his honour will be safe. The reasoning behind this is that a good knight wishes for battle to be celebrated and witnessed by Princes, Lords and other Knights "... by whose determination justice shall be remitted and not otherwise. And to do otherwise would be most indecent, outside of military discipline, and a custome belonging to common butchers, ruffians, and peasants..."

The need for this sort of legal justification for refusing to go into the woods and fighting an illegal duel demonstrates that people certainly did it, but also that there was a distaste amongst the nobility at the idea of their knights hacking lumps off each other without their permission. Fiore Dei Liberi in his instructional manuscript gives his fighting credential by saying that he was forced 5 times to fight with sharp swords and no armour, but that he would caution his students to never attempt such a thing. His reason being that when fighting without armour, one only needs to falter once to die, but an armoured fighter rarely looses his life when fighting in the lists, and as such Fiore would rather fight in the lists 3 times than attempt a fight without armour even once.

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The TL;DR is that honour duels absolutely happened in late medieval western Europe, they were supposed to be carried out under the sanction of an overlord, whose permission gave the battle similar legal standing to a just war, and the lord was the final arbiter who could decide when to end the fight. Despite this, men did fight outside the bounds of the law, often resorting to fighting without their armour. However, efforts were made to argue that this was unknightly behaviour.