How common was marriage between nobles and peasants in the Middle Ages?

There are some examples. King Erik XIV of Sweden married the servant Karin Månsdotter in 1567 (not really middle ages, but quite close). Karin Månsdotter was the daughter of a soldier/jailkeeper and a peasant. Only two of the six marriages Henry VIII made to secure an heir were with royal brides, and Elizabeth Woodville, queen of Edward IV of England, was also a commoner.

Since I have a great memory and access to the internet I quickly found a case of the child of a mighty noble and a peasant becoming his father's successor, but unfortunately they don't seem to have been married.

Oldrich, Duke of Bohemia 1012-1033 and 1033-1042 apparently had no surviving children with his unnamed wife, but had his son and heir with his mistress Bozena (d. 1052), daughter of Kresina. Bozena is said to have been a peasant. And Wikipedia claims that Bozena became Oldrich's second wife. Their son Duke Bretislav I (died 1055) was the ancestor of later generations of Bohemian rulers.

https://en.wikipedia.org/wiki/Bo%C5%BEena_(K%C5%99esinov%C3%A1)1

Fredegund (died 597), the notorius queen and widow of King Chilperic I, Frankish king of Soissons, was said to be of lowly birth.

Fredegund was born into a low-ranking family but gained power through her association with King Chilperic.1 Originally a servant of Chilperic's first wife Audovera, Fredegund won Chilperic's affection and persuaded him to put Audovera in a convent and divorce her. Gregory of Tours remarks that Fredegund brought with her a handsome dowry, incurring the immediate affection of King Chilperic.2

https://en.wikipedia.org/wiki/Fredegund2

Of course it is unknown if Fredegund's family was low enough in status to make her a peasant.

Emperor Romanus II (938/39-963), ruled 959-963, married 2nd, 957, Anastasia/Theofano, daughter of Krateros, a poor tavern keeper from the Sparta region of Greece, who was the mother of his three children. Of course it isn't known if her original home was rural enough for her to be called a peasant.

http://fmg.ac/Projects/MedLands/BYZANTIUM.htm#_Toc3595786323

https://en.wikipedia.org/wiki/Theophano_(10th_century)4

These are some examples I could think of and find of medieval nobles' wives of lowly birth, though two of them were wives of monarch's.

Probably more common than you would think.

Lots of nobles were little more than farmers with a coat of arms. Peasants could acquire a lot of wealth. Where the division between the two classes fell varied enormously between countries, as well as the relative status.

To give an example:

In Sweden, the noble class was created in 1280, when anyone who could equip and keep a war horse in the service of the King was declared to be exempt from other taxes. Thus, the entire nobility was created straight from the peasantry, so the difference between "peasant" and "noble" was not a matter of blood, but rather one of exactly how much money you had. Marrying someone slightly less rich than you would not be a big, deal.

Note that even if you want to restrict "nobility" to having some sort of conferred title, the only ones that really existed in Sweden were "duke" (almost exclusive for royals) and "knight", and a person like Bo Jonsson (Grip), who controlled two thirds of the country, never attained either (and he was, BTW, married to the daughter of a knight).

It's hard to prove a negative (outside of math, of course), but let me try to show why this would be extremely rare (and mostly exist in legends).

The institution of marriage exists to protect the woman and to seal a family alliance. Neither reason would apply to a marriage of a high-born and a low-born:

1. There is no alliance to seal (there is nothing the low-born's family can offer to the high-born family).
2. If a nobleman lusts for a low-born woman, she would, of course, need the protection of marriage, but there is no reason for the nobleman to provide it: he has enough power to take her by force or blackmail.
3. If a noblewoman lusts for a low-born man, she needs no protection; it is also an exceedingly rare situation for her to have a say in her life - nubile women were treated as assets to be traded to neighbors in marriage for an alliance.

Two other possible reasons are:

1. Give legitimacy to existing children
2. Religion

They require fairly specific combinations of circumstances which probably never arose together, e.g., a nobleman marrying a low-born girl (instead of taking her as a mistress) for religious reasons requires that he has high religious conformance but low social conformance.

PS. The above assumes "established feudalism". All royalty and nobility stems from warlords and their warriors and originally the boundary between high-born and low-born was relatively fluid.

PPS. Another aspect is the level of stratification. No duke has ever married a commoner, but many squires did.