The truth is it varies greatly depending on who the soldiers are fighting for and what area they are from. As the story also only revolves around the lords and ladies of the realm to we don't really hear what the lowborns use. There are some examples below of what people use but most of it relates to soldiers rather than people drafted into the army for wars.
They appear to use whatever they can find before marching off. The most common weapons being what they use for farming and daily work or what they can steal.
Ser Wylis and his brother Ser Wendel followed, leading their levies, near fifteen hundred men: some twenty-odd knights and as many squires, two hundred mounted lances, swordsmen, and freeriders, and the rest foot, armed with spears, pikes and tridents.
A Game of Thrones, Catelyn VIII
"Why, no older than your boy," Meribald replied. "Too young for such, in truth, but my brothers were all going, and I would not be left behind. Willam said I could be his squire, though Will was no knight, only a potboy armed with a kitchen knife he'd stolen from the inn. He died upon the Stepstones, and never struck a blow. It was fever did for him, and for my brother Robin. Owen died from a mace that split his head apart, and his friend Jon Pox was hanged for rape."
A Feast for Crows, Brienne V
This answer by @Aegon has a good description of standard issue for the Night's Watch so I won't repeat his quotes here only his summary.
Night's Watch has its own armory and armor. They forge their own weapons with the steel they get as a gift from Lords or the steel they purchase. Night's Watch provides the following to its members and recruits:
- Chain mails
- Boiled leather
- Padding made of wool
However, I have also found that they have spears and swords, longsowrds and bows and arrows.
He carried a heavy spear tipped in iron, taller than he was, and a sword hung at his side in a leather sheath. Across his chest was a gleaming black warhorn, banded with silver.
A Game of Thrones, Tyrion III
The sentries at the ringwall stood still in their footsteps, breath frosting and heads turned toward the west. As the sound of the horn faded, even the wind ceased to blow. Men rolled from their blankets and reached for spears and swordbelts, moving quietly, listening.
A Clash of Kings, Jon V
"I did not command it. I told you to do what needed to be done, and left you to decide what that would be." Qhorin stood and slid his longsword back into its scabbard. "When I want a mountain scaled, I call on Stonesnake. Should I need to put an arrow through the eye of some foe across a windy battlefield, I summon Squire Dalbridge. Ebben can make any man give up his secrets. To lead men you must know them, Jon Snow. I know more of you now than I did this morning."
A Clash of Kings, Jon VII
I'm not sure if it still stands today but Daemon Targaryen equipped the City Watch with the following equipment when he was made commander.
Governance bored this warrior prince, however. He did better when King Viserys made him commander of the City Watch. Finding the watchmen ill armed and clad in oddments and rags, Daemon equipped each man with dirk, short sword, and cudgel, armored them in black ringmail (with breastplates for the officers) and gave them long golden cloaks that they might wear with pride. Ever since, the men of the City Watch have been known as gold cloaks.
The Rogue Prince
Although it does appear now that at least some of the City Watchmen also have spears.
The Mud Gate was open, and a squad of City Watchmen stood under the portcullis in their golden cloaks, leaning on spears.
A Game of Thrones, Eddard VI
We get a good description of what the Lannister soldiers have, at least those with Jaime Lannister when he ambushes Ned.
Ned glimpsed ringmail over leather, gauntlets and greaves, steel helms with golden lions on the crests. Their cloaks clung to their backs, sodden with rain. He had no time to count, but there were ten at least, a line of them, on foot, blocking the street, with longswords and iron-tipped spears. "Behind!" he heard Wyl cry, and when he turned his horse, there were more in back of them, cutting off their retreat. Jory's sword came singing from its scabbard. "Make way or die!"
A Game of Thrones, Eddard IX
There's decent description given of the Lannister forces under the command of Gregor Clegane.
"Every man among them was mounted and mailed, my lord," Ser Karyl answered calmly. "They were armed with steel-tipped lances and longswords, with battle-axes for the butchering." He gestured toward one of the ragged survivors. "You. Yes, you, no one's going to hurt you. Tell the Hand what you told me."
A Game of Thrones, Eddard XI
Later when Tyrion get's the clansmen to the Lannister camp he requests the following weapons for them.
"First, though, I have some promises of my own to keep," he said as he sliced off a wedge. "I shall require three thousand helms and as many hauberks, plus swords, pikes, steel spearheads, maces, battle-axes, gauntlets, gorgets, greaves, breastplates, wagons to carry all this—"
A Game of Thrones, Tyrion VII
They essentially use what they can steal or take from their victims so are probably a decent representative of an everyday peasant.
There were no heralds, no banners, no horns nor drums, only the twang of bowstrings as Morrec and Lharys let fly, and suddenly the clansmen came thundering out of the dawn, lean dark men in boiled leather and mismatched armor, faces hidden behind barred half helms. In gloved hands were clutched all manner of weapons: longswords and lances and sharpened scythes, spiked clubs and daggers and heavy iron mauls. At their head rode a big man in a striped shadowskin cloak, armed with a two-handed greatsword.
A Game of Thrones, Tyrion IV
"And Shagga son of Dolf." That was the first voice, deep and deadly. A boulder shifted to their left, and stood, and became a man. Massive and slow and strong he seemed, dressed all in skins, with a club in his right hand and an axe in his left. He smashed them together as he lumbered closer.
Other voices called other names, Conn and Torrek and Jaggot and more that Tyrion forgot the instant he heard them; ten at least. A few had swords and knives; others brandished pitchforks and scythes and wooden spears. He waited until they were done shouting out their names before he gave them answer. "I am Tyrion son of Tywin, of the Clan Lannister, the Lions of the Rock. We will gladly pay you for the goat we ate."
A Game of Thrones, Tyrion VI
Jorah gives a small description of what the Targaryen forces under Rhaegar used when they met the rebellion at the Trident.
"Your brother Rhaegar brought as many men to the Trident," Ser Jorah admitted, "but of that number, no more than a tenth were knights. The rest were archers, freeriders, and foot soldiers armed with spears and pikes. When Rhaegar fell, many threw down their weapons and fled the field. How long do you imagine such a rabble would stand against the charge of forty thousand screamers howling for blood? How well would boiled leather jerkins and mailed shirts protect them when the arrows fall like rain?"
A Game of Thrones, Daenerys IV
We get a brief description of their equipment when the tale of the 3000 Unsullied is told to Dany.
"The Dothraki charged. The Unsullied locked their shields, lowered their spears, and stood firm. Against twenty thousand screamers with bells in their hair, they stood firm.
A Storm of Swords, Daenerys I
We also learn that they use shortsword too.
"Are all Westerosi pigs so ignorant?" he complained. "All the world knows that the Unsullied are masters of spear and shield and shortsword." He gave Dany a broad smile. "Tell her what she would know, slave, and be quick about it. The day is hot."
If the Unsullied felt the heat, however, they gave no hint of it. They could be made of brick themselves, the way they stand there. A thousand had been marched out of their barracks for her inspection; drawn up in ten ranks of one hundred before the fountain and its great bronze harpy, they stood stiffly at attention, their stony eyes fixed straight ahead. They wore nought but white linen clouts knotted about their loins, and conical bronze helms topped with a sharpened spike a foot tall. Kraznys had commanded them to lay down their spears and shields, and doff their swordbelts and quilted tunics, so the Queen of Westeros might better inspect the lean hardness of their bodies.
"They are chosen young, for size and speed and strength," the slave told her. "They begin their training at five. Every day they train from dawn to dusk, until they have mastered the shortsword, the shield, and the three spears. The training is most rigorous, Your Grace. Only one boy in three survives it. This is well known. Among the Unsullied it is said that on the day they win their spiked cap, the worst is done with, for no duty that will ever fall to them could be as hard as their training."
A Storm of Swords, Daenerys II