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The emergent gauge field characteristic of the Coulomb phase of spin icebetrays its existence via pinch points in the spin structure factor ${\cal{S}}$in reciprocal space which takes the form of a transverse projector ${\cal{P}}$at low temperature: ${\cal{S}} (q) \sim {\cal{P}} \sim q^2_\bot/q^2$. Wedevelop a theory which establishes the fate of the pinch points at low and hightemperature, for hard and soft spins, for short- and long-ranged (dipolar)interactions, as well as in the presence of disorder. We find that theirdetailed shape can be used to read off the relative sizes of entropic andmagnetic Coulomb interactions of monopoles in spin ice, and we resolve thequestion why pinch points have been experimentally observed forHo$_{1.7}$Y$_{0.3}$Ti$_2$O$_7$ even at high temperature in the presence ofstrong disorder.

Coulomb phase diagnostics as function of temperature, interaction range and disorder