It depends upon the device.
My Kobo mini has mainly books I've yet to read, though I do keep a few favorite series in their entirety on it.
My Sony has all of the books in my main Calibre library. Once read they no longer have the "New" icon, but I leave the read books on the device.
My Nexus has my entire main Calibre library, the undread books from my To Be Determined library (all new-to-me authors), as well as the unread books from my Vault library (which I made to put books I've already read and wanted to keep, but didn't feel I needed to keep on my Sony. Since its creation I've found it easier to keep the new-to-me books by the authors in the Vault library together, so though it started as a completety read library it now contains a mix of read/unread books by authors I basically don't consider favorites, but where I wanted to keep their books).
If I DNF a book or finish but didn't like a book, I delete it completely from my device(s) and Calibre. Unless it's a dud by one of my favorite authors, in which case I keep it because I want to have all of their books. My favorite author list has shrunk a lot over the years though, so there's not many books that fit this category.
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