I currently have a Mastercard, and am thinking of switching to
American Express, due to a better offer on a rewards program. 
However, I'm concerned about how widely accepted American Express is
compared to Mastercard and Visa.  My question is: how widely accepted
is American Express compared to Mastercard or Visa?  Please answer in
number of merchants, worldwide.  Also, I'm from Canada, so if there is
any indication of lower acceptance of AmEx outside of the US, that
would also be helpful to know.

As a free comment, American Express and Diner's Club are not as widely
accepted as Visa and Mastercard in shops, restaurants, smaller hotels
in Europe (where I live).  From my experience, I wouldn't travel with
just one card; often enough, a card will not be accepted for whatever
mysterious reasons by the vendor's call-in.  When you check into a
hotel for more than one night, sometimes it will block a larger amount
than the room  - hoping for your bar and restaurant patronage -  which
may upset other charges (at least this seems so).

The numbers below are from the companies' respective websites:

Visa: Accepted at 24+ million merchants in more than 160 countries. 
In addition to that, there are over 1 million ATM machines that accept
VISA cards.

MasterCard: Accepted at 23+ million merchants worldwide and does
business in 210 countries worldwide.  In addition to that there are
over 1 million ATM machines worldwide that accept MasterCard cards.

American Express: Accepted at several million merchants worldwide and
does business in 130 countries.  There are over 500,000 ATM machines
worldwide that accept American Express cards.