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In this article, we revisit classical length identities enjoyed by simpleclosed curves on hyperbolic surfaces. We state and prove rigidity of suchidentities over Teichm\"uller spaces. Due to this rigidity, simple closedcurves with few intersections are characterised on generic hyperbolic surfacesby their lengths.  As an application, we construct a meagre set $V$ in the Teichm\"uller spaceof a topological surface $S$, possibly of infinite type. Then the isometryclass of a (Nielsen-convex) hyperbolic structure on $S$ outside $V$ ischaracterised by its unmarked simple length spectrum. Namely, we show that thesimple length spectra can be used as moduli for generic hyperbolic surfaces. Inthe case of compact surfaces, an analogous result using length spectra wasobtained by Wolpert.

Simple length spectra as moduli for hyperbolic surfaces and rigidity of length identities