Do not just upgrade immediately. Take the time you need for an informed decision. And make a backup, of course. 🙂
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Read the changelog for the new version. Example for 3.4.1. The blog announcements and the Codex pages are not complete.
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Test the upgrade with an exact copy of your production site, including plugins and themes.
- Add, change and delete an user, a post, a term (tag, category).
- Log out, log in, test all active user roles.
- Test all custom JavaScripts. jQuery for example was added once so late in the pre-release process that plugin authors could not adjust their code in time.
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If you are not a developer and it is not a minor release (where just the third number changes) wait some days, look for the trouble-shooting thread in the wordpress.org forums. Example for 3.4. Minor security updates are usually safe.
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If you use the web interface to upgrade or SSH/FTP doesn’t really matter anymore.
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Plugin compatibility has to be tested. There is no other way to know how they play together.