I'm a PA, and my current CEO, and previous CEO, both had the same preferred method.
We have the inbox. We use tags for the emails -
Blue - Him to action
Purple - Me to action
Orange - No action, information only
Green
We then have one subfolder - Complete/No Further Action.
We each mark our action as complete, and once it is complete, it's moved to NFA. Orange emails which are NFA are left in his inbox for a week to allow him time to read, and once they're marked as read by him, I move to NFA folder.
Both of my most recent CEOs have disliked multiple subfolders, because often our emails could cover two subjects and it's down to person interpretation as to which folder it goes into. Also, if we hit slow wifi wherever we are, it can take a while for our laptop to perform a search in multiple subfolders if we're looking for something specific.
We work public sector, so everything is subject to FOIs, so we never delete emails. We just operate on the basis that, once one of us has marked as complete, it moves to NFA.
If he has emails marked for his action that are not marked as complete by Thursday, I send him an action list for Fridays to work through any outstanding emails.