macosmail.app
I have some filters set up in Gmail, for things like mailing lists. Messages skip the inbox, get tagged, and go straight to archive. This was not a problem in Mavericks, but now I'm getting the "new mail" noise all the time, when really it's a new message in the archive, from one of these filters, but nothing is new in my inbox.
The setting in my preferences is still: New message notifications: Inbox Only.
Any ideas? A bug in Mail?
I had a similar problem in Mail on Mountain Lion, where attachments did not show up for a message. When looking at the raw source (⌘⎇U) of the message, I also found that headers of the message were missing.
I managed to fix the issue by selecting the mailbox containing the broken email, and then using the "Rebuild" command from the "Mailbox" menu. After a short time, my emails reappeared, complete with attachments.
Mail does have an action called "Archive" in the Message menu that moves the current message in to a directory called "Archive." I use it all day every day on my work IMAP account just like in Gmail. I stash everything in there and just rely on search to find what I need. It works great.
I don't think there's a way to remap the Delete key to perform an Archive action instead, but Archive does have a keyboard shortcut: control-command-A. If you find that as clunky as I do, you can remap it in System Preferences' Keyboard pane.
You can also add Archive to your toolbar if that's more your thing.
Best Answer
My workaround…
Switch off the new message sound in Prefs > General
To the end of every rule, or at least the end of all possible paths, add a new Action - 'Stop Evaluating Rules'.
Add one more Rule at the end, like this…
Anything that gets that far through the Rule list will play the sound, anything diverted on the way will not…
...or you can selectively add that or any chosen sound at the bottom of earlier Rules, for instance if you wanted to differentiate or highlight any particular incoming path.