During the first council in the Rise of Tiamat the Council of Waterdeep deputizes the characters with a writ of emergency investigative powers.
While the module certainly gives us oportunities to fill in the gaps between adventures in the module, it did strike me as unusual that in no particular adventures within the module do these powers particularly come into play. It's not as if Old White Death, the Yuan-Ti, or the Cult of the Dragon would have any respect for the character's writ, and I don't see how these powers would particularly come into play with the Red Wizard or good dragons either.
How would you use these investigative powers in your game?
If the PCs have to do something illegal but not immoral (such as being out at night after curfew), the Writ acts as an automatic "Leave Us Alone" card.
If the PCs get into a fight within city limits and the police break it up, the PCs can show the Writ and it acts as a "Get Out of Jail Free" card.
If the PCs find something that requires Government support, the Writ can act as a +X in their favor for the Persuasion check with the magistrate / bureaucrat.
If the PCs show the Writ to the Red Wizards, the Wizards might decide the PCs are unwitting cats-paws of the Lords Alliance, not indepenent actors in their own right.
If the PCs really want to display the Writ before the forces of Evil and yell "Stop! In the name of the law!", well, let them ! 
They may have to take Old White Death down to the station for questioning...
https://www.youtube.com/watch?v=-8lDYrvTILc