Yes, I agree about pear cores. Never eat those either. Also, they tend to be a lot more sour than the rest of the pear.

And the stalks... I eat them when they're nice and fleshy, but with the apples we get, that happens very seldom. If it's just a dry twig, then I generally just throw it away (unless I'm incredibly bored and have nothing else to do than chew on a small twig).