The Osage Indians' Lands (Situational Irony)
After being forced to move on multiple occasions, the Osage finally settled in Osage County, Oklahoma. They were allowed to remain there, in large part, because the government viewed this land as a "big pile of rocks." This ends up having unintended consequences, as the Osage subsequently discover oil on this land and make a sizable profit from it. In this instance of situational irony, the Osage are allowed to settle land because it is perceived as worthless but then quickly find that it is extremely valuable.
Guardianship (Situational Irony)
After the Osage make money off of the oil beneath their land, they are quickly placed into guardian arrangements by the government. This is ostensibly to protect their finances, but everyone knows this is really to swindle them out of their money. These guardianships end up being extractive and abusive, leaving many Osage individuals either penniless or dead. In this example of situational irony, the "guardians" of the Osage turn out to be vicious grifters.
Hale Is Called King of Osage County (Situational Irony)
William Hale is often referred to as the "King of Osage County." He is respected in the community and holds a great deal of sway in local politics. He has a close relationship with Mollie and her family and plays an active role in the investigation of Anna and Rita's deaths. What is later revealed is that Hale is the mastermind behind the conspiracy against Mollie and her family and has been systematically killing them off to get his hands on their money. This terrible fact is an example of situational irony in that Hale is found to be the greatest enemy of people he claims to be the best friend of.
Hoover Suppresses Information (Situational Irony)
FBI Director J. Edgar Hoover aggressively controls the media portrayal of the Osage case, making certain that he and the Bureau are viewed in a positive light. It is extremely important to him that the case highlights the FBI as being honest and free of corruption. To do so, he covers up and downplays unflattering elements of the case, including the mishandling of evidence and the duplicity of various operatives. In essence, he lies to make the FBI look truthful. In this moment of situational irony, Hoover behaves dishonestly to make the Bureau look respectable and uncorrupt.