The main song that is mentioned is the Song of the Family. This doubles as both Kino's personal song and family anthem. It materializes itself throughout the story. The song can take on a happy rhythm or wailing tones. During peaceful moments, it speaks of safety, warmth, and wholeness. Juana also contributes to this family song by singing a variation of its theme using only three notes and endless varieties of intervals. She sings her own version of the Song of the Family to Coyotito (to comfort him), as he suffers through the effects of the scorpion's poison.

Despite its power, the Song of the Family can be drowned out by the Song of Evil, "the music of the enemy, of any foe of the family, a savage, secret, dangerous melody." The Song of Evil accosts the family when the scorpion first stings Coyotito. As Kino grabs the offending arachnid and grinds it into the ground, the Song of the Enemy roars in his ears.

Kino's people have always sung about "everything that happened or existed." So, when Kino fills his basket with oyster shells, he hears the Song of the Pearl That Might Be mingling with the Song of the Undersea.

Later, when Kino faces off with the trackers, the Song of the Family becomes a warrior anthem. It is "as fierce and sharp and feline as the snarl of a female puma." After Coyotito's death, Kino decides to return the great pearl to the sea. As he prepares to throw the pearl into the waters, he hears the Song of the Family as a battle cry. It is fierce and resolute, urging him towards his mission of returning the ill-fated pearl to the sea.

So, there are at least five songs mentioned in the story:

1)The Song of the Family and its variations.

2)The Song of Evil.

3)The Song of the Undersea.

4) The Song of the Pearl That Might Be.

5) The Song of the Enemy (which appears to be a variation of the Song of Evil).

This is, in reality, sort of a trick question. In chapter one of John Steinbecks' The Pearl the narrator states that all of the songs sung are considered part of the Whole. Therefore, if looking at the question from this perspective, only one song is sung. 

Another trick to this question is that Kino recognized life as songs. For example, the cooking of breakfast was a song too. Therefore, if looked at in this light, there are far too many songs to name because everything is a song. 

The first actual song mentioned in John Steinbecks' The Pearl appears in chapter one. Here, Juana sings "The Song of the Family." Very soon afterward, she begins to sing an "ancient song that had only three notes." 

Later, when the scorpion approaches Coyotito, the "Song of Evil"  and the "Song of the Enemy" are both heard.

Other songs heard in the novella are as follows.

-"The Song of the That Might Be"

-"The Song of the Undersea"

Therefore, there are three answers to the question: 1) too many songs to count (for everything is a song); 2) only one song (because everything is part of a single song); or 3) there are four songs (as named above).