I suspect that one of your WP 'approved' plugins is out of date or deprecated. This happens from time to time. As upgrades of WP come out, and the developer of a plugin doesn't keep up with it, then their code breaks, taking everything in site with it.
Do a plugin inventory. Go to Dashboard > Plugins > Installed Plugins in one browser window, and in another go to
Check each plugin page on WordPress.org and make sure of several things:
1. It's compatible up to WP 3.5.1 or 3.6
2. Make sure no one's citing it as broken.
3. Check the plugin's forum to see if others are complaining about their site's breaking or other calamities.
When you find any plugins that are cited as broken or having problems mentioned in their forum, or are not compatible, go back to the Installed Plugins list and deactivate it.
Then, go to Dashboard > Settings > Permalinks and without changing anything, save the settings.
Next, find a suitable replacement for any deactivated plugins and make sure it meets the requirements of compatibility and not being cited as broken or breaking things.
This will help immensely. I'd recommend to you (and your client) doing this each time WP updates, even with minor updates.