As a first point, two-handed swords and pollaxes are both the weapons of -dismounted- soldiers. In the later middle ages men at arms (knights and those who fought as knights) often fought on foot, so in these circumstances they would use these two-handed weapons. When mounted, the primary weapon of the man at arms remained the lance (backed up by a sword and/or a mace or warhammer), until men at arms were replaced completely on the battlefield in the mid to late 16th century.

The short answer is that it is hard to say exactly what dismounted men at arms are carrying into battle. Moreover, different weapons are used at different times and in different places; prior to the 15th or very late 14th century the pollaxe hand not yet been developed. In general evidence points to very long, strictly two handed swords being 'niche' weapons throughout the middle ages while pollaxes were the mainstay of dismounted men at arms in the 15th century. For the sake of simplicity, I will focus on the 15th century, and then speak about other periods of the later Middle ages in my conclusion.

Before going into detail about the form of two-handed swords, I should note that in general, swords are not armour penetrating weapons. Sword slashes are extremely bad at penetrating armour because they distribute the force of the blow through the entire length where the blade contacts the target. Compared with axes or hammers, their lower point of balance gives them less leverage when they hit. Because of this, Alan Williams tests showed that sword blows delivery much less force than is needed to penetrate plate armour or mail armour (unless you hit a weak point). Swords points are considerably more effective than blades, but primarily when they are thrust through gaps in armour rather than through the armour itself. One way to do this was 'half swording' - gripping the sword halfway up the blade and using the extra control to find a gap in the armour to thrust into.

For the purposes of this post, I will define a 'two handed sword' as a sword that can -only- be used in two hands. Generally these have a long grip and have a blade over 40 inches long (four feet would be long even for these, though). These are a contrast to 'longswords' or 'hand and a half swords' with blades generally around 36 inches - these swords can be used in two hands, and are much easier to use two-handed, but can be used in one hand in a pinch, which is useful, especially on horseback. This ability to be wielded in one or both hands, mounted or on foot, makes them much more versatile than the larger 'true' two-handed swords, and they are much more common. It is hard to tell if men at arms would go into battle armed just with a longsword; perhaps some did, but our evidence (the large number of other weapons in inventories among other things) points to them mostly being used as secondary weapons - useful, but not the first choice. EDIT: Some form of sword (hand and a half or one handed) would have been on the hip of most of not all men at arms, so they were ubiquitous even if they weren't 'primary' weapons.

By contrast, the specialized two-handed sword is fairly rare. Though many people characterize them as 16th century weapons, there are a number of examples that survive from earlier, so we can say with certainty that two-handed swords were used in the Middle Ages.

There are illustrations of very large swords in some 13th century manuscripts, while some surviving 13th century ‘Great Swords of War’ are nearly 40 inches in length - which is approaching the length of a true two-handed sword. Froissart records one such two-handed sword being use by the canon de Robesart in a battle in 1358: "He held a sword of two hands, with which he dealt blows so great that none dared to face them." Some exceptionally large 14th century swords survive, including one with an overall length of 6ft 8 1/4 inches hung over the tomb of Edward III in St George’s Chapel, Westminster Abbey. This is likely a ceremonial ‘bearing’ sword - its weight and length would make it unwieldy on the battlefield.

By the 15th century, however, we have a number of surviving 2-handed swords including some particular swords with English and Danish associations. This Sword at the Wallace collection is from around 1450. It has a hollow-ground blade of roughly diamond section. With a blade over 46 inches long, this is a true two-handed sword. At 6 lbs, it is heavy but manageable. Its pommel and guard are characteristic of English swords found in the Thames and near the 1453 Chastillon battlefield, and similar swords are depicted on English Funerary effigies. There are also a number of Scandanavian two-handed swords, which feature a distinctive, very long partially wire-bound grip and a blade of between 36" and a little over 40". Some of these have a ricasso, an unsharpened base of the blade, effectively making the grip even longer. You can call these either short two-handed swords or very long-gripped 'hand and a half' swords.

The 16th century is the period from which we have the most surviving two-handed swords; the most famous two-handed swords, the German ‘bidenhanders’/’zweihanders’ are 16th century weapons. Though some surviving swords are so large that they may have been ceremonial (swords that are very heavy are particularly suspicious), a decent number of these swords are functional weapons such as this one at the Wallace Collection. This sword's blade is just under 4 feet, and it has a ricasso which is partially protected by spurs that emerge from the blade. These are more prominent in later 16th century two handed swords. Some later 16th century two handed swords are over four feet long, it is sometimes hard to tell which are for parades and which are for battle. Based on what we know of soldiers armaments, these swords were less common than halberds (let alone pikes, which become the dominant infantry close-combat weapon) on the battlefield. These swords would probably have been carried by a partially armoured common infantryman rather than a fully armoured man at arms.

While the two-handed sword is a bit of a niche weapon for its entire history, and the longsword/hand-and-a-half sword is a useful all-around backup, the pollaxe is an extremely effective and yet versatile weapon for men fighting on foot. Its combination of hammer head, spike and axe blade means that it can deliver concussive blows, cleave with the axe and punch through armour with its spikes. The effectiveness of pollaxes against armoured men can be seen gentleman being wounded in 'friendly' pollaxe contests, like Sir Pandolpho Malatesta being wounded by the Earl of Warwick in the 15th century. Suggestively, many of the skulls recovered from the mass graves at Towton have square holes in them, the exact shape of the four-sided spike of a pollaxe (or warhammer). Compared with other polearms like bills and halberds, it tends to be a bit shorter and perhaps more balanced. As such they were both effective on the battlefield and surprisingly agile weapons for foot combats in tournaments - and they were very popular in both.

In an early period, before the pollaxe was developed, the preferred weapon might be simpler two-handed axes - there are accounts in Froissart's chronicle of the Earl James Douglas using one in the 14th century - he 'took a two-handed axe and plunged into the fight... none was so well protected by helmet or late as not to fear the blows he dealt.' In England and France in the 14th century, medium-length spears or 'short lances' were used by dismounted men at arms in large numbers - the armoury of the Tower of London had hundreds of short lances in inventory in the later 14th century.

So if we distinguish between 'hand and a half' longswords, 'true' two handed swords and pollaxes, we might say that longswords were versatile weapons that were often used as secondary weapons on the battlefield by both mounted and dismounted troops, 'true' two-handed swords were 'niche' weapons, and pollaxes were perhaps the dominant weapon of dismounted men at arms on 15th century western European battlefields, but in other periods a different weapon would be most common.

Sources:

Tobias Capwell - Masterpieces of European Arms and Armour at the Wallace Collection - this includes details about the 15th century sword at the Wallace Collection, the polearms noted above, and the use of two-handed swords in the Middle ages, including the translation of the Canon de Robesart passage in Froissart.

Edge and Paddock - Arms and Armour of the Medieval Knight

The Knight and the Blast Furnace, Dr Allan Williams

Records of the Medieval Sword, Ewart Oakshott