On my phone I have the default search set up as Google, but I try to use the Bing search button whenever I can. Sometimes I'm forced to use Google because I can't seem to do advanced searches with Bing--like I can't limit the time period I'd like to search. Also the Bing client in Windows Phone doesn't understand things like "time in Melbourne" whereas if I were to type that on Bing in a desktop browser, it'd give me the actual time in Melbourne. Also, for really complicated searches, I just use Google.
On my computer, all my defaults are set to Bing. If I need Google, I'll type it in manually. I find it strange that Bing will only let me choose a time period of up to one month, while Google will let me choose a time period of up to one year, or a custom period. In addition, Bing does not seem to recognize the Philippine Peso (PHP) as a currency, while Google does just fine, so I use Google for currency conversions.
I really want to use Bing as much as possible, but there are some limitations that Bing has that Google does not. It's annoying sometimes.