How much evidence (if any) do we have of the sword-and-dagger style being used on the battlefield? The only instance I can recall off the top of my head is Smythe's injuction that pikemen should be able to draw and fight with their swords and daggers simultaneously, but this evidence is basically theoretical in nature and I don't know whether this suggestion was ever really put to practice in a massed combat situation.

Can anybody help me with this? At the moment I'm leaning towards the conclusion that sword-and-dagger is simply not a battlefield swordsmanship style, but I'd be glad if anybody can prove me wrong.

Unfortunately I cannot give you any specific references but I heard from many sources that rondel was a typical part of the weapon's collection a knight would carry into a battle.
When you close in with your opponent in the grapple distance, you cannot use your sword very effectively because there is no space to perform a proper thrust or a cut. Sure, you can pommel them in the face or go for the disarm or simply break the distance, but rondel gives you an option to decisively finish a battle right there.

Arms&Armor website says about rondel: "The side arm of the knights who fought the battles of Crecy, Poitiers, and Agincourt. " Now I believe, the rondel that they pictured there is a civilian version. The military rondel had a heavier triangle blade and did not have any sharp edges - it was only a trusting weapon that had a good chance of penetrating riveted chain mail.

I doubt Smythe's instructions came from nowhere. We know various pikemen carried both swords and dagger. Are we sure they didn't use them together?

Another example is Francisco Pizarro fighting with sword and dagger at the Battle of Cajamarca. Of course, that was more of a massacre than a battle.

I would think that daggers would be used for extremely close-in fighting, as in, when it's too close to use a sword (such as two men in plate armour that are grappling, for instance). They would therefore be an essential piece of military equipment, but I have no evidence to support or deny the theory of using daggers and swords simultaneously save my own experiments.

I have tried using a medieval-style sword with a dagger, and the dagger's reach is tiny compared to that of a sword. It would seem mainly useful for parrying, as indeed daggers sometimes were when dueling with rapiers. If you're going to be in a battlefield situation and need something to parry with, why not just use a shield?

It is my gut feeling that the sword/dagger combination would be a backup in the event that the shield strapping or shield broke. I suspect shields broke more then any other weapon. Getting a replacement shield in the battle might be hard. My first instinct would be to look around on the ground for a shield or a second sword since I can fight with either hand. But even if a person has not trained two-sword they can still block cuts with it. Essentially they would be using it as shield. If their are no swords to be found, then pulling a dagger would be the only option left. I still think it might be better to block with the sword, which has more mass, and stab the face with the dagger. I say face, because I don't think a dagger will even go through leather, so you need an unarmored area. Sorry, I have no historical basis for this. Its just what I would do if my shield broke.

In a civilian conflict with rapiers, daggers are great parry weapon. You use it as a windshield wiper parrying away from your body and you can thrust with it if you get in close.