My answer is going to focus mainly on the early middle ages, and my main area of knowledge here is northern Europe, but what I say will have applied more broadly in time and space.

The average medieval peasant would have received no formal training in singing. However, singing would have been a big part of everyday life. Children would learn work songs from their parents for activities such as ploughing, milking, spinning, and churning. Milking songs are attested in Ireland as early as the 9th century:

Another method of stimulating a cow's milk flow is by singing or playing music during the milking. Our sources contain references to music increasing cows' milk-yield, though it must be admitted that they occur in literary contexts and therefore cannot be taken as firm evidence of actual practice. According to the ninth-century tale Longes mac nUislenn the singing of the sons of Uisliu was so melodious that it increased the milk-yield of cows and other animals by two-thirds. Another tale attributes the same rate of increase to the cows, sheep and other milking animals which hear the flute of Mac Díchóeme. (Fergus Kelly, Early Irish Farming, Dublin 2009, p. 39).

Although, as Kelly points out, those references are literary, milking songs were still sung in the Gàidhealtachd in the 20th century so it is likely that they were actually sung in the period the literary texts were written. After all, the stories rely on an association between music and increased milk yield, the same principle that underlies milking songs in modern Scottish Gaelic. You can hear an ethnographic example of a Scottish milking song here: http://tobarandualchais.co.uk/en/fullrecord/9259 (See also Margaret Bennett, "'A Song for Every Cow She Milked…': Sharing the Work and Sharing the Voices in Gaeldom", The Phenomenon of Singing, 6 (2007), pp. 35-47.)

Any peasant with a natural talent would have been able to sing these songs well - they'd certainly have the practice, given how hard they worked! Dance songs, while not very well-attested in the earliest periods, were also a feature of later medieval peasants' lives. For example, the word "carol" originally referred to a song sung during circle dances. If the examples of late modern dance and work songs are anything to go by (a regrettable anachronism, but music of the common people was not well-documented in early times and there were no recording devices), these songs usually did not have harmonies. Instead, it was common for one person to sing the verses and everyone to join in on the refrain. (If you want to see examples of late modern Gaelic dance songs, check out Will Lamb's 2012 edition of Keith Norman Macdonald's Puirt-à-beul, or listen to one here: http://tobarandualchais.co.uk/en/fullrecord/34863.) Harmonies are most strongly associated with the development of polyphonic church chant. The origins of polyphony are disputed, but the earliest solid examples are from the 9th century, the Musica enchiriadis and Scolica enchiriadis. You can hear a recreation of 11th century polyphony from the Winchester Troper here: https://www.youtube.com/watch?v=vGJvzJvqo_c

Most medieval peasants did not become priests, monks or nuns, so they would not have received the musical training and education needed to read chant notations. The religious services they attended would sometimes provide the opportunity to sing hymns, a practice that goes back to Biblical times. While most hymns and psalms would be sung in Latin, there were religious songs sung at least partially in the vernacular, such as the Irish litanies (see the entry "Medieval Ireland: liturgical song" in The Companion to Irish Traditional Music, ed. by Fintan Vallely (Cork 2011), pp. 449-450). People who regularly attended church might learn to sing the hymns, even the Latin ones, provided they had the inclination.

We also know from Einhard's Life of Charlemagne that lay congregations were expected to sing during parts of the liturgy:

He [Charlemagne] made careful reforms in the way in which the psalms were chanted and the lessons read. He was himself quite an expert at both of these exercises, but he never read the lesson in public and he would sing only with the rest of the congregation and in a low voice. (quoted from the Penguin Classic Two Lives of Charlemagne).

Of course, Charlemagne and many of his court could understand Latin (Einhard says that the king spoke it "as fluently as his own tongue"), but it does suggest that when the congregation were capable of singing along in church, there were opportunities for them to do so. Monks and priests would have been singing and chanting many times throughout the day, so any peasant who found himself taking up an ecclesiastical order would have even more experience singing than other people. However, it's disputed how often peasants would be able to take up such an order. Similarly, in many European cultures there were designated secular singers, but these were usually elite ranks not available to peasants, such as the bardic classes in Ireland and Scotland or the Scandinavian skalds.

All of this is to say that between church, work, and social dances, the average medieval peasant would have had many opportunities to hone their vocal abilities - more than the average person today. However, the rarefied education needed to sing complex harmonies and other more advanced techniques would not have been available to the average peasant.