The medieval period was more than 1000 years long, and covered whole Europe, parts of M.East and Asia, and the military culture in it varied wildly, so I can only give a very general TLDR answer, unless you specify what century you are interested in.

Simply put, there was a great spectrum of "quality" between a common foot soldier (presumably a peasant volounter or a drafted farmer/townsman) and knight. In between there would be various groups of more and less skilled soldiers: ad hoc volounteers, volounteers on short contract, hired brigands, mercenaries, swornmen, squires and knights, as well as higher nobility.

Usually, the higher up the feudal chain the person was, the richer they would be, meaning they would own better armor and weapons, and have had more time to train.

Realistically however, the best trained group would be elite mercenaries, and if said band of mercenaries were famous enough, they could be feared. A good example of that would be the leftovers of the Sirotci Bratctvo (the Orphans) an ofshot of the Hussite Army turned mercenaries. While not exactly feared, they were considered a significant force to be reconed with, especially since the Hussite Army defeated almost every force it fought against, including the most elite knights that the German Empire could muster against them. Some of the orphans were knights (or to be more precise, former-knights, outcasts), but most were commoners, and their prefered tactics included soldiers on foot, armed with polearms, crosbowmen and armored carts.

Prior to that, the most prolific "feared" warriors in early Medieval times were viking mercenaries, hired by almost all European powers of that time (included, but not limited to, Poland, Kiev Russia, Byzantine Empire, knigdom of Wessex etc).

Technically, the prowess of a warrior in battle was not based on his personal skill, but how well he cooperated with others, and contrary to Holywood depictions, medieval battles were not a riot of personal duels, but combat of two or more tightly packed forces of men shielding and protecting one another, with differently armed soldiers taking different roles (example: the viking shiledmen protected the spearmen, who in turn cooperated with the axmen and all of them protected the archers, as well as provided a temprary stop for riders).

In such a formation, a good tactician or a leader could possibly stand out, but individual warriors would be of much less importance.

The situation would be different with riders, who often WERE famous, and "allowed to show off" with pre battle duels and feats of provess, and thus, would be feared for their skill. You however, asked specifically for footmen.

Your question is very broad since the Middle Ages is huge but the height of medieval warfare on foot was in the Later Middle Ages so I'll be addressing that. It's very difficult to get into the mind of the common medieval soldier so it is hard to know to what extent they would have feared noble warriors. I'll be avoiding using the term Knight here because in the later Middle Ages it took on some important connotations and significance so not all fighting nobility had to be a knight and not all knights necessarily fought. There is also a rather extensive scholarly debate on infantry warfare during this period so it's possible some of what I say will conflict with someone else's opinion, that happens.

The classic narrative of the rise of infantry warfare began with the Scots in the 13th century. This is the period of the first Scottish War of Independence, William Wallace era. Scottish nobility were actually well known for fighting on foot during this period primarily because Scotland was relatively poor, war horses were expensive, and Scotland was not geographically well suited to the sort of large war horse used in medieval warfare at the time. They had light cavalry but never really invested in the heavy cavalry of other kingdoms. These nobles would stand in among the rest of the men and fight on foot rather than as a separate mounted division. At Bannockburn in 1314 King Robert I actually fought on foot at the front of his army. There's a great story from the battle of a rather hot-headed young English noble charging the King's formation and Robert side stepped his lance thrust while cleaving his head open with an axe. Unfortunately this broke Robert's weapon and he had to retreat to the back of the formation to get a new weapon.

There are a couple of significant battles between Bannockburn and Crecy but I'll be skipping over them. Courtrai is an interesting battle for foot soldiers but the Flemish who won weren't nobles by any measure. At Crecy we essentially see the English emulating the tactics of the Scots with their soldiers primarily fighting on foot, including the nobility, only they had added a substantial supporting group of medieval archers which did wonders at ruining the French's formations. The traditional narrative is then that this tactic, with some variation, was employed by the English with great success from Crecy through to Agincourt in 1416. There's some truth to that and some problems with it but I'm not sure the nuances are in relation to your question so I'll leave them for now.

One of the reasons I brought up these two particular battles is they also cover an important change in how English armies were formed. Unfortunately I'm not as sure on how other Kingdoms' ran their military. At the time of Edward I, the father of Edward II who lost Bannockburn and also noted villain in Braveheart, English armies were primarily formed of soldiers who showed up as a result of feudal summons. Essentially the King's lords, the earls and counts and such, owed the King military service in exchange for the lands he had granted them (well...usually his ancestor had granted them to their ancestor but you get the point) but this didn't actually mean that only the Earl in question had to show up. Depending on the size of the land grant the Earl might be obliged to come along himself and to bring with him several sergeants at arms, usually cavalry, as well as a set number of foot soldiers and archers. Bigger plots of land meant more soldiers had to come. This obligation essentially passed down the chain a bit since an Earl with a large holding of land would in turn have divided it up to more lesser nobility. So the Earl would still have to bring some men with him from the lands he personally oversaw and he'd have to make sure that his vassals showed up to the King's summons. This system was understandably really impractical but it mostly worked up until the end of Edward I's reign where it all began to begin to give under it's own weight for a lot of reasons I'll spare you now. This is important for your question because during the time this was common practice (primarily from the reigns of John through to Edward I, although it was used to some extent before) the common soldier would likely have been a peasant.

During Edward I's Scottish wars a few alternative methods of recruiting armies began to change. One thing Edward I did was instead of forcing his nobles to come along out of Feudal obligation, which often entailed Edward paying their expenses and providing food which had proven extremely costly, he began to ask his nobles to come voluntarily at their own expense and to bring soldiers with them. This might sound like a bad deal but one important difference was that while serving under Feudal obligation any treasure acquired by taking a city or ransom had to be shared with the king, while serving voluntarily you got to keep your own loot. This meant that if you were successful you could make a lot of money but if you lost in battle you were out a lot of cash. The other trend that happened here was more nobles began paying money instead of showing up for feudal obligation. There was technically a fine if you didn't show but nobles began just paying the fine up front instead of showing and kings, like Edward, could use the money to hire mercenaries. This generally meant that from the late 13th century and on you could expect a higher caliber of soldier certainly in Medieval English armies and they would likely be better equipped, experienced and less afraid of a noble warrior.

That ended up a lot longer and more rambling than I had initially intended. Answering your question is very hard because it's so broad, I can try and answer more specific questions when I get up tomorrow if you'd like!

Some sources: Michael Prestwich Armies and Warfare in the Middle Ages

Michael Prestwich The Three Edwards

Marc Morris A Great and Terrible King

Geoffrey Barrow Robert Bruce

Kelly DeVries Infantry Warfare in the Early Fourteenth Century

It's really important to understand that your question is framed in a way that implies a fundamental misunderstanding of how combat (at least on a large scale) has gone down in the period contemplated by the terms you use -- "men-at-arms" and "knights" suggest the Medieval periods, a huge swathe of time, to be sure...

When you say things like: "knights on foot," and "to be avoided in combat," this implies a sense that people were fighting in a sort of pell-mell, choose-off-and-fight, individual way in battle. This was almost never the case in battles of any scale whatsoever. Regardless of time period or place, whether it's Vikings, Byzantines, Swiss, or Genovese, units fought in formation, using shield walls, cavalry groups, pike squares, lines of crowssbowmen, etc. The idea was that you fought as part of a unit, in cohesion with other individual soldiers, against units of other soldiers, also in cohesion. Breaking off from that unit and showing down against some dude, armor or otherwise, mano e mano, in the midst of battle, was a ticket to getting surrounded and cut down tout de freakin' suite. You see this kind of thing happen a lot in movies -- battles breaking down into a melee where one or more main characters does something heroic to someone else -- but in real life, once that happened on any sort of scale, the battle was basically over.

So, to answer your question, while it was indeed possible (and in the later Medieval period, not uncommon) for cavaliers to dismount and engage on foot using weapons specifically designed for this purpose, especially pole-arms, it was not really possible to just wade into combat and pick somebody out, Braveheart style, and go after him. Or, conversely, to "avoid" someone dangerous. You got into formation, and you went where your commander said to go. Sometimes that meant standing just outside of spear range of another unit for minutes on end, shouting at them, then suddenly charging for just a few seconds and pulling back. Sometimes it meant firing a volley of arrows at people so far away you could barely see the shape of their bodies. But whatever you did, you did it with comrades, barring exceptional circumstances.

So, probably not the answer you were looking for, but /u/Valkine gives a pretty good breakdown already of some of the advantages (and disadvantages) nobles had in single combat versus your average dude, so start with his comment for that angle.

A knight, even on foot, would have some of the best arms, armour and training at the time. A the battle of Agincourt the English knights fought on foot due to a lack of horses, it had been a long, hard campaign through France, and due to a need to concentrate their forces in anticipation of the French charge.

Crusaders in the holy land would also be forced to fight on foot at times, espcially if they had lost their horses from disease or fighting. During one battle the Hospitallers lost an incredible amount of horses to Saracen raids on their rear guard, and it was only a final desparate charge led by the head of their order which saved them. (The details are mentioned in Bennetts "Fight techniques of the medieval world", forgive me for not having the details to hand.)

While Heavy Cavalry were one of the most effective weapons on the medieval battlefield, a knight of foot was still an incredible effective and deadly warrior.

As to your second question, there wouldn't really be a way to "avoid" fighting anyone one the lines of battles met. Levi with little training and poor equipment would be butchered and swiftly seen off by Knights, and Knight would no doubt be insulted to not be faced against their opposite.

Movies portray battles a seires of dynamic duels. In truth it would be a crush, a true bloody melee where one side breaks and runs, and the other side runs them down.