By my definition, dead end links are any links that don't help. These would include links to pages that no longer exist on your site, that perhaps need a 301 redirect, links from websites that have nothing to do with your keyword topics, or links from websites that have really bad DA on pages with bad PA.
Webmaster Tools within Google is probably a great place to start. You can see where Google is finding errors by looking under Crawl > Crawl Errors. There you will find pages that might have been deleted without a redirect and are therefore showing a 400 Error. Also you can see who is linking to your website under Search Traffic > Links To Your Site. That's probably a really strong place to start.
OSE only gives you a sample of the pages that are pointing links to your site, but you could find those domains there as well. Majestic SEO offers a free account with some Referring Domain information as well.
As far as fixing the issues, well if ts a link on a website you don't want to be linked from, you can start by emailing them and asking for it removed. If its a link to a page that has been deleted or renamed, just 301 redirect that page to a live, working page so that when its clicked through, the user ends up on a working page.
Google's number one goal is user experience. If their users click on a link from Google that ends up on a page that doesn't exist, it works against you. If Google sees a link on a referring domain that is pointing to a page that doesn't exist, its a similar situation. If Google sees two pages with the same content, that's another notch against your site. That's where Canonical links come into play.
It's a long process so just hang in there and make sure the powers at be know how important it is because its going to take a large amount of man hours to fix this issue. Can you check Webmaster Tools and tell me how many errors it finds under Crawl Errors and how many links are under Total Links and under "Who links the most" check domains that have a ton of links to your site that might not be helpful to your site.