I would not use Lastpass. I manage my passwords locally using KeePass software.
keepass seems to be windows-only

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I'm not currently doing this, but "for real", I would probably use an encrypted /home/joe folder anyway, so I could just put a text-file with the login pws there.
However: lastpass (or any other password store that has a browser addon) is very convenient.
What are the specific problems with using lastpass?