1Password, and it's been awesome. If you use the non-hosted version you control the whole thing. LAN only sync, or Dropbox, Rsync, however you want to sync it if it matters. Otherwise they've got a hosted version which can also give you web access in a pinch and handle all the syncing for you. I have been with them for long enough that my only option was a non-hosted version and I keep in sync with Dropbox and it works wonderfully.
The Mac client is very nice to use. The Windows client works well but isn't as nice looking. The browser extension is awesome- AND you can put MFA keys in there as well.
EDIT: And I forgot to mention the main reason I chose 1Password so long ago. Their iOS app is awesome and their mobile Safari integration works quite well.
No direct Linux support but you can absolutely run the windows version 4 in Wine. It's not without a few crashy glitches but for just using stored passwords it works fine.
It would be very nice if they did support Linux and people have been asking for it, but there is a passable workaround- and frankly one I'm willing to work with because it works so great everywhere else.
When I tried it out I wanted 1password family so I could share some accounts with my partner. 1password 4 doesn't support their cloud datastore; the version that does will not run on Wine.
However, I could use the webapp on linux. It was a bit annoying but I could have dealt with it. The other complaint I had was the UX for Android. Having to switch my keyboard every time I wanted to enter a password was very annoying. Hopefully that gets better with the recently announced Autofill API for Android O.
The Mac client is very nice to use. The Windows client works well but isn't as nice looking. The browser extension is awesome- AND you can put MFA keys in there as well.
EDIT: And I forgot to mention the main reason I chose 1Password so long ago. Their iOS app is awesome and their mobile Safari integration works quite well.