The following query will return all posts that are associated with more than one category. This query could be useful in the scenario where you would want to migrate a site to a category structure where each post is only assigned to one category. If this is your goal, you may want to check out my Radio Categories Plugin
$posts = $wpdb->get_results( " SELECT p.*, COUNT( p.`ID` ) AS 'category_count' FROM $wpdb->posts AS p, $wpdb->term_taxonomy as tt, $wpdb->term_relationships as tr WHERE tt.`taxonomy` = 'category' AND tt.`term_taxonomy_id` = tr.`term_taxonomy_id` AND tr.`object_id` = p.ID AND p.`post_type` = 'post' GROUP BY p.`ID` HAVING COUNT( p.`ID` ) > 1 ORDER BY COUNT( p.`ID` ) DESC " );


