A point x in a normed space X is said to be singular for a given tiling of
X whenever each neighborhood of x intersects infinitely many tiles. We show
that, when X is infinite-dimensional and all tiles are convex, special poi
nts in the boundary of tiles (like extreme points or PC points, if any) mus
t be singular. Under the further assumptions that X is separable and doesn'
t contain c(0), singular points abound among the smooth points of any bound
ed tile. Finally, in any normed space a tiling is constructed which is free
of singular points and whose members are both bounded and star-shaped; thi
s disproves the conjecture that Corson's theorem might apply to star-shaped
bounded coverings.