POP 4EVR!
I need a local, searchable backup/history of ALL my work email, going back years, and I need it in MY control, not some service provider. POP.
My Mac (Mail app) is the master repository (and the one with extra spam-handling), and once I get a message there it's deleted from the server so my iPhone/iPad inbox can stay lean.
iPhone/iPad can check for new mail (manually works best for me) so I'm not reliant on the Mac for seeing new messages; but deleting from mobile doesn't delete from server, it just keeps my mobile inbox lean.
(Outgoing mobile emails get BCC'd to me, and the Mac Mail app automatically routes them silently to my local Sent archive.)
In short: once I got things set up nicely, I can do what ever I want to manage my email with my iPhone/iPad (keep a lot, chuck it all, whatever), and it doesn't matter: the Mac still has a searchable history of EVERYTHING, without me having to think about it.
I also set my "other Macs" up the same way as my iPhone/iPad. They're not the "master copy," but they can view and send with no harm done.
And luckily, I haven't had a problem with iOS 9. (Only received 1 attachment that I can think of, though.)