...was Agnes singing just before she first meets Dr Livingston? Part of the Catholic mass known as the "Gloria". The particular one she was singing is the Ambrosian "Gloria" (named for St. Ambrose) and is an example of Ambrosian, as opposed to Gregorian, chant. This is characterized by mostly one note for most of the line and the elaborate ending of each line.
...was Sister Genevieve and her family singing just after she took her vows? "Le Bon Vent" (The Good Wind), a traditional French song.
...did Agnes sing in the courtroom? "Over the River Charlie", a traditional song about a girl who loses her virginity. Agnes sings only the first view lines before she is led out of the courtroom, i.e., "Charlie's neat, and Charlie's sweet, and Charlie, he's a dandy / Every time he goes to town, he gets his girl some candy. / Over the river, and through the trees, over the river to Charlie's / Over the river and through the trees, to bake a cake for Charlie." The full text can be found here.
...did Agnes sing just before the credits? That's "Pie Jesu" from the Requiem Mass. Lyrics are "Agnus Dei, qui tollis peccata mundi, miserere nobis / Agnus Dei, qui tollis peccata mundi, miserere nobis / Agnus Dei, qui tollis peccata mundi, dona nobis pacem." (Translation: "Lamb of God, who takest away the sins of the world, have mercy upon us. / Lamb of God, who takest away the sins of the world, have mercy upon us. / Lamb of God, who takest away the sins of the world, grant us peace.")