My favorite local coffee shop takes Discover but not Amex. Their rival does the opposite.
One advantage of Amex is that many places that DO take Amex, ONLY take Amex.
I have *never* run into a place that only took Discover. It's always the 3rd or 4th wheel to Visa/MC/maybe Amex.
Still, Visa definitely has the most extensive network these days. I wouldn't want to go through life without at least one Visa card.