In the Continuation of John Fordun's Scotichronicon (c. 1440), Walter Bowers notes the popularity of Robin Hood stories among the masses; he may or may not be referring to the rich tradition of Robin Hood ballads in Britain, some of which still survived in oral tradition last century.

There may be some other examples out there that I have never come across, but it's generally difficult to say much before the early modern period. In the last few centuries, there have been songs ubiquitous throughout English-speaking vernacular song cultures in North America and Great Britain (Barbara Allen, for instance) – although I would hesitate to say throughout the English-speaking world. Would this have been the case in the medieval period? Possibly, but I don’t think it likely until the establishment of the broadside ballad trade in the sixteenth century, which provided means of mass distribution.

Similarly, the representation of vernacular song in A Song of Ice and Fire is absolutely not realistic. GRRM treats songs like we might treat them today -- they have a set tune and a set text, and any deviation from these is in error. In Westeros, an almost entirely oral cultural with no culture of mass print or musical recording, there would be significant variance between different singers' versions, keyed into their own artistry, their source material, and their memory.

We can see dramatic variations in items from the same song family in the material collected from singers of Anglo-American "folk" singers over the last several centuries -- and these traditions have been often mediated through print culture (despite the characterizations of Victorian and post-Victorian scholars), which acts to homogenize texts. But as in medieval England, there’s no broadside trade in the Guttenbergless Westeros. Furthermore, mobile Gipsy and Traveller communities (and their kind) have played key roles in spreading songs in recent centuries – I’m not sure Westeros has an equivalent of this, either.

Without these mediating influences, it's not particularly likely that The Bear and the Maiden Fair would be ubiquitous throughout Westeros, especially not in a form that would allow someone from the Reach and someone from Winterfell (or even two strangers who only live a town apart) to sing together without first comparing notes.

As always with questions about the Middle Ages, we are limited in what we can say about popular music by the sources available to us (though medievalists are constantly finding new things). Most obviously, there was not an accurate enough notation for preserving secular music until roughly the 13th century, so we’re at a loss to talk about the popular musical culture earlier than that with any great confidence.

Setting aside that caution, there were clearly popular songs that circulated regionally if not internationally. The famous 12th-century philosopher-theologian Abelard was as famous for his singing as his thinking. He seems to have been somewhat of a “rock star” in Paris, where he was celebrated for his popular songs, especially love songs, none of which survive.

The earliest notated secular song in English is [“Sumer is icumen in”] (https://www.youtube.com/watch?v=sMCA9nYnLWo) (Summer has come) from c. 1250. [In the manuscript in which it survives] (http://www.bl.uk/learning/images/medieval/artmusicliterature/large96674.html), it is written above a Latin Easter hymn, with which it shares its melody (which came first is unsettled). Likewise, the hymn “Angelus ad Virginem” survives in both its [original Latin form] (https://www.youtube.com/watch?v=-ith5FF-P9U) and in a [Middle English translation] (https://www.youtube.com/watch?v=zyCCQQ94ktQ). First composed in c. 1290, perhaps by Franciscans, it was well enough known that Chaucer could reference it in his “Prioress’s Tale” and assume his audience knew it. (It’s the account of the angel Gabriel’s Annunciation” to Mary of the conception of Jesus.)

Probably many popular songs did what “Angelus ad Virginem” did: take a church melody and adapt it for secular use. But this was a 2-way street: the composer Guillaume Dufay took the melody of the popular, anonymous secular song [“L’homme arme”] (https://www.youtube.com/watch?v=KHNMgDIk1QI) (the armed man) and used it as the foundation for [one of his masses] (https://www.youtube.com/watch?v=fLwMEBlBBB4), written c. 1450. Popular songs got preserved locally like the [“Coventry Carol”] (https://www.youtube.com/watch?v=iRmO8cnRkOQ&list=PLsCQ5hYBQ6MNkkxOE8l82nwHlIGNY_JSC&index=12) (still heard at Christmas), first mentioned in 1392 but not preserved until 1529. There are lots of [other surviving English carols] (https://www.youtube.com/watch?v=pB5x3dFepn0&list=PLsCQ5hYBQ6MNkkxOE8l82nwHlIGNY_JSC) from the later Middle Ages, including the famous [“Agincourt Carol”] (https://www.youtube.com/watch?v=SW6EhketxYo) that Apocsurvivor mentions.

It’s safe to assume that minstrels and troubadours, who moved around towns and castles performing, helped spread popular secular music. Some of these were probably courtly songs, meant only for the upper classes. An example would be the lament Richard the Lionheart wrote while in captivity pleading for his barons to raise the money to ransom him: [“Ja nus hons pris”] (https://www.youtube.com/watch?v=ELeXBomZACg). There are other well-known secular songs connected with the crusades, such as Guiot de Dijon’s [“Chanterai por mon corage”] (https://www.youtube.com/watch?v=KOi2Hd49OWE). The famous Carmina burana manuscript (early 13th century) preserves dozen of [popular songs] (https://www.youtube.com/watch?v=4tCQd1TKsHg). And so on, and so on.

Cynthia Cyrus has a well-informed online [Introduction to Medieval Music] (http://www.the-orb.net/encyclop/culture/music/orbmusic.html#secular) that discusses medieval secular music (and church music too).

Edit: proofreading

Related question: isn't it true that they used to recycle a lot of tunes (and put different words to them), and that many national anthems were based off of the tunes old drinking songs?

Furthermore, were songs like 'The Rains of Castamere' real? As in were songs commissioned to pau tribute to something in particular?

There is the Lochamer-Liederbuch, although it is German it is full of popular songs for Lute and voice with recognisable tunes. It might be a good idea to note that many of the tunes that are used in Lutheran hymns were originally popular (sometimes quite saucy) songs. The point of them was to use tunes that everyone would know and thus make it easy for everyone to participate.