IMO Les Mis corners the market here, between Eponine's On My Own and Fantine's I Dreamed a Dream.and Empty Chairs at Empty Tables, as well as Valjean's Bring Him Home. In addition...
-- What Would You Do? from Cabaret is heartbreaking, particularly when sung by Lotte Lenya
-- All I ask of You from Phantom, specifically as sung by the Phantom at the end of the show;
-- To me, Move On is very sad in the context of the show; young George is at the end of his rope; all I could think of was how alone he was at that point, and the song is so haunting.
-- Rose'sTurn. It may be bravura, but it is very sad. At that point, she is questioning her whole life.
-- Not While I'm Around, in the context of the show.
-- When Your Feet Don't Touch the Ground from Finding Neverland.
-- Make Them Hear You from Ragtime
-- This Nearly Was Mine from South Pacific
-- I've Grown Accustomed to Her Face from MFL
-- The Party's Over from Bells Are Ringing
-- I Never Loved You from Cyrano
-- Once Upon a Time from All American
-- Send in the Clowns from ALNM
-- If He Walked Into My Life Today
-- Some Other Day from On The Town
-- The Impossible Dream, specifically in the finale
-- Far From the Home I Love and Anatevka from Fiddler
-- Something Wonderful from TKAI (it is a beautiful song, and always makes me sad when it is well done)
-- For Good From Wicked
-- Stay Alive and Who Lives, Who Dies, Who TellsYour Story from Hamilton
Agree with posters re You'll Never Walk Alone, Tell Me It's Not True, Words Fail, I'd Give My Life For You, Welcome Home, I Won't Send Roses.