  For a given elliptic curve E in short Weierstrass form, we show that almost
all quadratic twists E_D have no integral points, as D ranges over square-free
integers ordered by size. Our result is conditional on a weak form of the
Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses a
correspondence of Mordell and the reduction theory of binary quartic forms in
order to transfer the problem to counting rational points of bounded height on
a certain singular cubic surface, together with extensive use of cancellation
in character sum estimates, drawn from Heath-Brown's analysis of Selmer group
statistics for the congruent number curve.
