Her rendition of the Alfred Noyes poem "The Highwayman".
"Dante's Prayer" is another one.
Then there is "Ce He Mise le Ulaingt/The Two Trees". The lyrics are from a William Butler Yeats poem — but the singer's rendering of some of the bleak imagery, plus the beautiful melancholic pipe intro, is incredibly isolating.
"The Lady of Shalott" is another one. The whole song is beautifully rendered, gorgeously played, and poignant — but it's the last verse that can really do it because of a) the shift in the melody to bring the song to a close, ending exactly where your ear and heart wants it to and b) the true tragedy of the story — that if Elaine had come to Lancelot and professed her love, rather than simply pining away for him, the whole sordid mess with Guinevere, Arthur's death, and the fall of Camelot would never have happened.
And especially "The Dark Night of the Soul".
A lot of her songs have elements of this, since many are based on old poems (or specifically, Irish ballads) which were tragedies — "Coventry Carol", "The English Ladye and the Knight", and Shakespearean passages set to music such as "Cymbeline" and "Prospero's Speech" (the high, ethereal voice, almost a descant, combined with the melancholy pipes and strings on that last one really does it). But there are also some surprise examples, such as her early piece "Standing Stones" (the moment where the lover's scream "disturbed the silent night" is especially visceral due to her forceful delivery of the line, followed by quite the wail of pipes) and "Skellig", the story of a dying monk passing on his knowledge and books to a younger brother of the order. Her latest album is in fact chock full of these: "The Sally Gardens", the title track "The Wind That Shakes the Barley", "The Death of Queen Jane", and "The Parting Glass"—not only sung as wistfully and mournfully as possible, but actually ended with the last line broken and choked out, followed by one last, gasping breathas if the singer had just died.