Are we sitting comfortably?

Alot of infantry did indeed carry backup weapons, mostly knives of varying lengths, some carried falchions, English archers carried mauls (often knives too) as a multipurpose tool for driving stakes and breaking up earth and as a weapon against armoured opponents (technically they didn't carry 'swords' even though they carried what could effectively be used as a sword ie: falchions and long knives, because swords were the weapon of knights, the nobles wouldn't have allowed a commoner to carry such a weapon nor would a soldier not encased in heavier armour encumber himself with such a weapon). Its a common misconception that infantry becomes utterly innefective when confronted with cavalry and without spears or once the cavalry has broken through a wall of spears, accounts suggest that infantry with long daggers and the like would harrass the horses and cause alot of disruption, unhorsed men could then be stabbed on the ground through gaps in their plate armour (if they were rich enough to own some, it wasn't as common as people think), the Welsh and Irish spearmen were known for their ability to damage cavalry with spear and knife.

Halberdiers and Billmen were used in much the same way as spearmen because both weapons had a thrusting point, their particular advantage was that they were still effective against armour without using the force of a charging mounted man against himself because of their heavy chopping blades and rear-facing spikes, again these troops would likely have carried backup weapons, images of billmen often show them with a buckler and small sword attatched to the belt by leather thongs. Bills also had the advantage against cavalrymen because of their curved hooks which could unseat a rider, they were also used against foot troops to hook behind a leg.

Two-handed swords were often used by dismounted knights, and they, like poleaxes were intended for breaking plate armour when it was more common during the later medieval period. I'm unfamiliar with the 'Zwei - Hander' soldiers of the HRE historically, although I do know that the Landschnekt mercenaries did employ men with zweihanders known as Doppelsoldners (double-pay-men) who were payed double for the risky job of assaulting pike formations. This illustrates the fact that while zweihanders were useful for breaking up enemy formations they were still vulnerable against pikes, as most things were by the end of the fifteenth century. After a certain stage pikes pretty much dominated the battlefield because of the successes of the Swiss and Landschnekts.

Its important to remember that alot of non-missile infantry (and alot of missile, in fact) in medieval armies was either dismounted men-at-arms (knights,squires,sergeants) or professional mercenaries. In the case of dismounted men this means that to some extent the cavalry and infantry arms are the same men and deploy differently depending on circumstances, at Agincourt the dismounted French knights are reported to have broken off sizx feet of their lances to use as a polearm and this occuring there and likely at other places lead to the development of weapons for use by dismounted knights such as the poleaxe.

Missile weapons were most effective against anyone wearing armour 'below' plate in terms of its protective qualities and many melee weapons evolved out of a need to counter cavalry and infantry as well as break heavier armour, hence the evolution of the bill, poleaxe, zweihander, halberd, mace and warhammer, axes were also used. Knives and estocs (thrusting swords) were used for stabbing between gaps and visors.
Non-spear infantry fits in because it could be kept to the rear or flanks of spear or pike formations and engage when necessary, cavalry often also operated in this manner.

That ended up kinda meandering all over the place because I lost my train of thought but I hope that helps a bit. If anyone could add to clarify a few things that would be cool.