Spade's distrust of power is obvious in his resistance to the police and district attorney's actions and opinions of Archer's murder and subsequent events. He's also obviously correct, when you consider the DA's crackpot theories. Anyone and everyone who tries to move him in any direction he views with mistrust, and resists.
He views everything anyone tells him as a potential lie, and in the few incidents in which he temporarily believes someone he catches on pretty quickly that his initial distrust was correct. The facts about the Falcon he only believes after he has carefully sifted through every available piece of information to seperate the truth from the lies.
This lack of trust in the mental acuity of others, and disbelief in anything they tell him, is based on his past experiences as an investigator, and as a soldier. There is an implied sense than his experience in the First World War soured him on believing what authority figures tell him is true. His distrust is absolutely central to his character, and he is proven right at every turn in the story.