People that are experiencing the 404 errors, I may have tracked down the issue:
In your panel.dreamhost.com > Manage Domains > Edit…
Do you have Extra Web Security? checked on?
If so, uncheck it, and click the Change settings button.
Wait ten minutes or so, and see if the 404 errors go away.
If the 404’s are gone, change that setting on all of your WordPress sites.
Second possibly, if the above doesn’t fix it: On the same page, set PHP Mode to: PHP 5.6 CGI (not FastCGI) — this one may be particularly helpful for sites with a lot of plugins.
Save and wait for ten minutes or so.
Let me know if one or both of the above helps.