I think that the layout/design should always use SSIs wherever possible...it means that to update the design only one page need change.
Also for PHP sites I always put the database connection information in an SSI file..
Hope that helps!
From my point of view, SSI isn't client-side, but since I'm not certain where this should go I'll leave it here until someone says otherwise.
Since SSI requires server time, I would limit the SSI to code that would best be served by a SSI script, such as session and access information. Otherwise, you may notice slower response time.
So, what sort of information do you intend to relegate to the SSI?
James' statement about one-page updates is true; however, the same may be said for js and css files (both of which are loaded and executed by the surfer's computer and, therefore, client-side processing).