  A survey is given of the work on strong regularity for uniform algebras over
the last thirty years, and some new results are proved, including the
following. Let A be a uniform algebra on a compact space X and let E be the set
of all those points x of X such that A is not strongly regular at x. If E has
no non-empty, perfect subsets then A is normal, and X is the character space of
A. If X is either the interval or the circle and E is meagre with no non-empty,
perfect subsets then A is trivial. These results extend Wilken's work from
1969. It is also shown that every separable Banach function algebra which has
character space equal to either the interval or the circle and which has a
countably-generated ideal lattice is uniformly dense in the algebra of all
continuous functions.
