  We study the relationship between asymptotic characteristic initial data at
past null infinity and the regularity of solutions at future null infinity for
the massless linear spin-s field equations on Minkowski space. By
quantitatively controlling the solutions on a causal rectangle reaching the
conformal boundary, we relate the (generically singular) behaviour of the
solutions near past null infinity, future null infinity, and spatial infinity.
Our analysis uses Friedrich's cylinder at spatial infinity together with a
careful Gr\"onwall-type estimate that does not degenerate at the intersection
of null infinity and the cylinder (the so-called critical sets).
