One way you can find the article with the most shares is through AddThis analytics. You can login to addthis.com and view the analytics or you can get shares by content or the top shared content through an API:
http://support.addthis.com/customer/portal/articles/381264-addthis-analytics-api#.Uu0jP3ddWTM
And then lastly, if you want to know when shares occur in real-time, there's an event api that you can use to listen for a share and then do an ajax call to your own server to track the share:
http://support.addthis.com/customer/portal/articles/1365497-addthis-javascript-events#.Uu0jmHddWTM
And look for the addthis.menu.share event. Hope this helps.