I'd call "Thank Goodness" from 'Wicked' a regret song. It doesn't start out that way, but it sure ends up that way. Also "No Good Thing" from the same show.
Of course "Empty Chairs at Empty Tables" from Les Miserables is all about regret and sacrifice, so that also fits well. Also "How Could I Ever Know" from 'The Secret Garden'.
"Heartbreaker" from Bright Star really hits on the sacrifice aspects (realizing he can never see the love of his life again, for he can't bear to hurt her).
I can't believe nobody has mentioned "Telephone Wire"!!!! That was such a painful song of regret. Specially when she sang
"Telephone wire, Stop! Too fast! Telephone wire, Make this not the past. This car ride, This is where it has to happen! There must be some other chances. There's a moment I'm forgetting Where you tell me you see me....
Say something! Talk to me! Say something! Anything! At the light, At the light.
This can't be our last......"
What's more painful also about this song is how she couldn't accept the fact that that was it. And that she was trying to remember sometHing more than what had actually happened. But that was it. No more telephone wires. They came to the end of the journey and he wasn't able to tell her what she wanted him to say. Its so hard to regret the things you can never ever change.