How to determine (via php) if site is using a static home page?

get_option( 'page_on_front' );

If this value is not 0, there is a static page.

Additional note from Johannes Pille:

get_option( 'show_on_front' );

… returns post or page. But be aware of custom post types as front page.

A good way to find common options is to visit the (hidden) page wp-admin/options.php.