For arbitrarily small values of varepsilon >0, we formulate and analyse the Maxwell system of equations of electromagnetism on varepsilon -periodic sets S^varepsilon subset {{mathbb {R}}}^3. Assuming that a family of Borel measures mu ^varepsilon , such that mathrm{supp}(mu ^varepsilon )=S^varepsilon , is obtained by varepsilon -contraction of a fixed 1-periodic measure mu , and for right-hand sides f^varepsilon in L^2({mathbb {R}}^3, dmu ^varepsilon ), we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic “singular structures”, when mu is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for L^2 functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method.