Anger drives a lot of the action in Martin McDonagh’s “Three Billboards Outside Ebbing, Missouri,” a movie filled with scenes of violence and notes of despair that lands in a deep and invigorating way because of its forgiving soul.

The rich human tapestry on display in this fictional Missouri town centers on the efforts of distraught Mildred Hayes (Frances McDormand) to goad beloved police chief Bill Willoughby (Woody Harrelson) into picking up the pace in the investigation of her daughter’s rape and murder.

Mildred, a tough and fearless woman, sets Ebbing into an uproar and onto a journey toward something approaching self-examination when she erects the eponymous billboards with their pointed message: “Raped while dying. And still no arrests? How come, Chief Willoughby?”

Onstage and onscreen, McDonagh has specialized in the art of pitch-black comedy with a strong moral underpinning; his heightened, operatic sensibilities allow for a degree of cleverness and absurdism that mixes with strong emotional intelligence.

In “Three Billboards,” as with “In Bruges” and others, McDonagh’s characters and their actions don’t conform to familiar behavioral conventions. They can be despicable and kindhearted; calm and furious; driven to reckless actions at some moments and totally methodical at others.

Even with scenes of Molotov cocktails being tossed and men being thrown out of second-story windows, the movie maintains its dramatic focus on these inner lives. The filmmaker and his cast masterfully capture them, the unspoken thoughts and feelings that complicate our understanding of why they carry themselves the way that they do.

McDormand reveals the fractured soul cleaving away at Mildred, who is typically seen glowering in a jumpsuit and affecting an aura of having no time for anyone’s nonsense, but is in fact lost in a state of unending sorrow. Harrelson’s Willoughby seems confident and poised, but he’s grappling with his own mortality in a real and unexpected way.

The most affecting performance in the movie, the one that best represents what makes it great, comes from Sam Rockwell. The veteran character actor plays deputy officer Jason Dixon, a man prone to bursts of extreme anger and violent aggression, who is widely hated for his misconduct and looked down upon by seemingly everyone in town but Willoughby as being hideously unfit for his position. But you understand him; you see him alone with his domineering mother, you appreciate the desperation with which he seeks to please his boss and do things the right way.

McDonagh makes sure we appreciate the full scope of this man’s existence, and it is his story that illustrates the idea at the heart of “Three Billboards Outside Ebbing, Missouri,” which is that no one is beyond redemption.