Mildred's motivation
The driving event behind the movie is Mildred's placement of 3 billboards regarding her daughter's murder, to presumably put pressure on the police to solve the murder. Mildred is basically portrayed as the "good guy" in the movie, for whom we are rooting. She is eccentric, but she is not a nut; she is very intelligent. So one thing I don't get is why she would place these billboards at all. The police chief (Woody) explained all of this extremely well in their encounter. All of the leads went nowhere. The DNA did not match anybody. Mildred suggested that he get DNA from everybody in the town, but the chief correctly explained that he can't do this. He further explained in the movie that these murders are usually solved when the killer confesses in a bar or in a jail cell, and that is completely true. Many crimes are not solved by the pro-active investigation by the police, but rather by the perpetrator running his mouth (or doing something else stupid, such as returning to the scene of the crime). Mildred does not ever really demonstrate that the police are neglecting their duties in any specific sense.
I want to pull for Mildred, but this basic premise underlying the movie just seems off. This would work if in fact the police were not doing their job, but every indication in the movie is that the chief does care about solving the crime and has done everything he can. So if the chief is not really a bad guy, and is not neglecting the investigation, then why place the billboards?
If the billboards were focused on the killer (and not the sheriff), for the purpose of getting a tipster to come forward or to keep the crime in the news, that would make sense; but attacking the chief just does not seem to make sense. I understand that Mildred would be in great pain following the loss of her daughter; I got that. And those things can make people act irrationally. But it seems to be flaw in the structure of the movie.