Well, about everyone around that time who even thought of himself as a soldier or warrior had some kind of sword. Pikemen, arquebusieurs, bandits, knights, you name it - everybody. Halberds and similar polearms were very popular as "close support" weapons in the pike formations, being rather better armor-smashers than pikes and rather more manageable at close distances (naturally depending on the lenght of the shaft, which in turn depended on the weapon's intended use - the ones carried mainly for anti-cavalry work were naturally long), and the older two-handed swords developed into the infamous greatswords, great for hacking through pike shafts and their carriers. Later on those became the weapon of important folks' bodyguards, carried partly for appereance and intimidation but also for their all-around efficiency against most any other weapons.

The Spanish sword-and-buckler men were a leftover from the Reconquista, whose last stages had involved a lot of fighting in mountains and fortifications. They turned out to work really well as close-support shock troops in pike squares, and also carved a bloody swath through the American natives (obsidian sword-clubs and cotton armor are at a sort of disadvantage against steel helmets, breastplates and Toledo cut-and-thrust swords...).