  The structure factor for electric field correlations in the two dimensional
Coulomb fluid is simulated and compared to theories of the dielectric function.
Singular changes in the structure factor occur at the BKT insulator to
conductor transition, as well as at a higher temperature correlation transition
between a poor electrolyte and perturbed Debye-H\"uckel fluid. Structure
factors are found to differ in the canonical and grand canonical ensembles,
with the poor electrolyte showing full ensemble inequivalence. We identify
mechanisms of `underscreening' and `pinch point' scattering that are relevant
to experiments on ionic liquids and artificial spin ice respectively.
