I dug up this question in a AH thread from 3 years ago but it seems like most people were focusing on details with sharpness of an executioners sword. That is, in my opinion, not a good example. It would be like a question in 500 years asking "how accurate were old gunpowder based kinetic energy firearms?" and people answering using longest ranged confirmed kills.

The best insight we can have into how sharp European weapons, among them swords, are the Battles of Visby in 1361 and Towton in 1461, smack dab in your requested timeline. This gives a good viewpoint as, well, the evidence isn't some concocted story but real forensic evidence that archeologists have only been able to come across due to the employment of mass graves. While I fully acknowledge it's impossible in some instances to know if it's a sword, axe, halberd or some other polearm or weapon, it's more of an indication in the quality of the refined metal and ability to refine an edge and sharpen said metal. Some of the evidence is graphic so please be warned.

The Battle of Visby was a pitched battle between mostly farmers and conscripts lined up against a mostly mercenary force led by the Danish King Valdemar IV. Here is a presentation from the Swedish History Museum that details a select number of physical evidence. It's impossible to know exact armament types of the Danish and Gnutish soldiers, but it's likely the Danish mercenaries would be armed with swords. Swords were more expensive than axes but were generally lighter and much more versatile than an axe, given a sword can slash, parry and stab. The injuries on the bones were likely a result of a tactic employed against an armored opponent wielding a shield but one who doesn't have heavy armor on their lower extremities. SIDENOTE: Visby is a fascinating look into how a peasant levied army could be armored. The average farmer that comprised the Gotland Defense Force was surprisingly well armored. However, this could just be a product of Visby being a major hub of the Hanseatic League, which would give better access to trade goods available to those on the periphery.

The Battle of Towton was in the English Civil War and is actually the bloodiest battle (known) to have happened on the British Isles. [The University of Bradford has a good writeup on this battle with some archeological evidence scoured through.] (http://www.bradford.ac.uk/research/faculties/life-sciences/areas-centres/biological-anthropology-research-centre/projects/towton-mass-grave-project/). Most of the injuries present at Towton follow a pattern indicating they're inflicted by swords.

As with Visby, it's impossible to know if the damage was done by a sword or a halberd. Regardless, the physical evidence that exists certainly details that whatever weapons used were plenty well sharp to clear through armor, muscle and bone mass or some combination of those 3. The sheer animosity of the wounds at Towton (many done after death would have rendered the victim inert) indicates these were done as a form of defacement and out of animosity.

Also, note on your sentence that swords would be bashed against armor and other weapons. This wasn't the case with the sword. The sword is not a breaching weapon unless against a lightly armored opponent or lightly armed area on the opponent (face, neck, shoulder, elbow, knee, etc). The sword, as per manuals and physical evidence, would be employed as to what Visby shows. This is why Western swords remained straight, for versatility. It allows the wielder the option to stab into a gap in armor or slash onto the unarmored foot of a careless farmer hiding behind a shield. The main focus of the European sword still remained at a piercing weapon.

There are cases of faithfully reproduced swords chopping through chainmail on a slash but this was only to test the capability. The impact usually rolled or chipped the edge, depending on the hardness factor of the steel, which would happen with any sword.

Source

Battle and Bloodshed: The Medieval World at War by Lorna Bleach and Keira Borrill