In the Sunless Citadel module, from Tales from the Yawning Portal, Passive Perception is mentioned about half a dozen times, but Passive Investigation isn’t mentioned once.
However, both passives are increased by +5 by the Observant feat, and both are included when it comes to the Inquisitive rogue subclass in XGtE. In most mechanical and class mentions of Passive Perception vs. Passive Investigation, it seems they are supposed to be equals.
Passive Perception is used as the defensive stat for enemies attempting to use Stealth, but no such claim is made for Passive Investigation.
So when is Passive Investigation used, beyond being an automatic permanent minimum roll for an unmade active investigation check?
Personally, I like to use them as reactionary/defensive stats that the DM rolls against, for things that a character can miss (somebody passing by, a drop of water falling from the ceiling, knowing the intended target of a shot fired from an assassin). This is not, however, an official use of the passive skills. I’m interested in regular uses of the passive skills for the sake of this question.