I see a lot of stuff about how Spears are pound for pound the strongest melee weapon out there, save for specific circumstances. What I have to wonder is if that is the case why do swords and axes and such all seem so prevalent in history as opposed to everybody just working on better spears and going to town?
What are the advantages of spears?
One-handed spears: Reach, can be used with shield.
Two-handed spears: Reach and/or speed. Short two-handed spears typically have more reach than one-handed weapons other than spears, and more reach than many two-handed non-spear weapons, and are fast (due to the leverage of the two-handed grip). Very long two-handed spears (i.e., pikes) lose much of this speed but have tremendous reach.
The general advantage of most spears is reach. When is reach less useful? In a close tangled melee. Of course, you can shorten your grip on a two-handed spear and use it at close range, but you no longer have the advantage of reach. Also, the length of the weapon means it can be difficult to move it around freely in a crowded melee. A solution is simple: carry a shorter weapon as a sidearm, something that is easier to use in a confined space. A sword is very convenient, since it can be worn at the side, with its scabbard protecting you from its edge, but still being easy to draw at need.
Similarly, a sword is a very convenient sidearm for archers, crossbowmen, musketeers, etc. If they don't carry a sword as a sidearm, at least some other short weapon will be useful to carry.
Thus, swords can be very common on the battlefield, even if nobody is carrying them as their main weapon. Off the battlefield, swords are still useful, as they can usually still be worn at the side while carrying out one's day-to-day activities, if the law allows. If the need for defence or the display of status or masculinity demands the wearing of weapons, a sword is much more convenient than a spear.
Typically, the sword is not a competitor vs the spear as far as being carried/worn goes, but a companion. One type of weapon that is a competitor is the two-handed cutting polearm, such as the halberd. These are typically shorter, heavier, and better at cutting. Their weight and their length allows them to deliver cutting blows that should be effective against at least the lighter-armoured portions of armoured soldiers (arm and leg armour was often under 1mm thick, vs the arrow-proof 2-3mm often seen for breastplates and helmets). Against unarmoured opponents, halberds etc. won't provide any great advantage (but they allow more flexibility in angles of attacks, and better hooking of shields, legs, etc.), and the reach of spears can make spears the better choice. Against well-armoured enemies, the halberd is useful, and can either be used to supplement spearmen/pikemen or as a substitute for them (Flodden Field (1513), was largely a battle of bill (essentially, a halberd with a concave-edged hooked blade) vs pike, with the bill-using English winning).