Minimal steps to reproduce:
1. Install Drupal core.
2. Go to admin/config/development/performance, turn on page caching and set the maximum expiration of cached pages to a nonzero number (e.g. a few minutes).
3. Log out and visit the home page using Safari (make sure to visit e.g. http://example.com rather than http://example.com/node).
4. Log in via the block on the left sidebar.
5. Click on the site logo which will take you back to e.g. http://example.com. Notice you will appear to be logged out.
If you refresh the page in your browser it will correctly show you as logged in again, though.
What's happening seems to be that Safari is having issues with the Cache-Control max-age setting combined with the Vary: Cookie setting and is serving pages from the browser cache when it shouldn't.
For more details and previous discussion, see: http://groups.drupal.org/node/191453
This is not reproducible with other browsers and is quite likely to be a Safari bug: https://bugs.webkit.org/show_bug.cgi?id=71509
However, it's possible Drupal is doing something wrong here, or at least there is something we could be doing in Drupal to work around it.... Anyone have any ideas?