Something haunts your adventurer. Maybe he made a vow he didn’t keep. Maybe he was defeated in an honorable fight and ordered the death of his foe anyway for some greater principle. Maybe he isn’t actually cowardly but once ran from a battle and left others to die.
As a Minor Hindrance, the shameful circumstances aren’t generally known—it just haunts the hero. He might go out of his way, against all reason, not to repeat the mistake. Or he might give in to the same set of circumstances and hate himself for it.
As a Major Hindrance, his deed is well-known— or at least known among those he cares about. The other player characters should be told the tale as soon as possible (preferably at character creation). If not, it should be revealed by nonplayer characters at some point, and occasionally used against the hero.