Looking at the Failings in the Rules, at Disgraces in particular (p. 73)

...at least some small portion of his pride was proven false. ...Make an unexpected sacrifice....regrets having made that decision... eats away at his soul and causes him to sometimes act in ways to make up for his past failings.

So If I roll Destruction for Disgrace, is Destruction something that I REGRET or is it the way I act sometimes to make up for past failings?!

If it is something that I regret, what ways should I act to make up that I killed a bunch of civilians and blow up their house?

The only Disgrace that automatically represents a Regret is the Disgrace of Regret. So the Disgrace of Destruction basically means your character disgraces himself by capitalizing on "Opportunities to spread suffering, by any means." They may not regret the Disgrace itself (the Betrayal-types probably don't) but they may regret the fallout the Disgrace causes.

But I don't know. The description of individual Disgraces suggest they are ongoing things, but the general "Failings" section describes them as things your character dislikes about himself. Who knows.

I'd personally just say the only Disgrace you automatically regret is Regret.