In what cases are you able to start a postdoc without a "conferred" PhD? Many PhD programs only confer their degrees one, two or three times per year. If you  successfully defend and submit your dissertation after the scheduled conferral date, you must wait until the next one to get the PhD. At many of these schools you can ask for an official document that basically states that all requirements for the PhD have been successfully completed and the degree will be conferred on the next conferral date.

Do postdoctoral research positions accept this type of official document in lieu of a PhD, to begin work, if a PhD is required? Is this specific from position to position or country to country? I will be receiving a PhD in the US and looking for postdocs in either the US, Canada, Europe, Japan or Australia.

Note this question is less about applying to postdocs and more about officially starting them. In other words, assume you have accepted a post doc offer, and the job requires a PhD, would having an official letter from the University stating the dissertation is done and everyone has signed off suffice to start?
If the postdoc requires "degree in hand" by a certain date, then there's little flexibility. The actual degree must be conferred by the university.

That being said, a good number of postdocs instead require that all of the requirements for the PhD be completed before a certain date. This usually means that that the dissertation has been deposited, if not defended.  On the extreme end of flexibility, some will simply take the word of the dissertation advisor that the dissertation will be finished by the start date.

Some postdocs will even allow you to start without a degree in hand if all of the requirements for the degree have been satisfied. Some will be satisfied with a letter from the advisor, others will require a letter from the registrar. How strict they will be may be correlated to how many times they've been burned in the past with non-completers.  

I don't think there are stats are on how many postdocs (and in which fields) are strict vs. flexible. What I think you'll find is that once a flexible department has been burned by a "postdoc" who never finished their PhD, then they shift to being more strict.

tl;dr: Ask the postdoc program if they are willing to be flexible and if so, which documentation they require.