Canonical URLs and domains to most SEOs are a common issue they run across. How each search engine handles the various patterns of URLs that seem or are duplicate to each other, may differ.
In a Bing Community thread, there is a comment from Brett Yount, a Program Manager at Bing Webmaster Center, on how Bing typically handles WWW vs non-WWW issues. By that I mean, if you both the http://example.com/ and http://www.example.com/ URLs return the same page, without redirection. Brett said:
It really doesn't matter if the site your link is residing on is www or non-www. More important is the structure of the URL they are using to link to you--especially if your site is not canonicalized using 301 redirects. When not redirected, we treat the non-www and the www version of your site as two different sites, so if you have links out there with and without the www, your whole site rank is affected. Canonicalizing your site to one or the other will automatically cause us to combine all the links and apply them to the chosen version of your site and may help increase your site rank.
Forum discussion at Bing Community.