Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent formal power series in the Planck constant and its inverse. These are conjecturally captured by the Nekrasov-Shatashvili and standard topological string free energies, respectively, via the TS/ST correspondence. Based on numerical evidence in the examples of local P2 and local F0, we conjecture that their resurgent structures involve peacock patterns of singularities and infinitely many rational Stokes constants, while we obtain an analytic prediction on the asymptotic behavior of the fermionic spectral traces in a WKB double-scaling regime dual to the standard 't Hooft limit. We solve exactly the resurgent structures of the spectral trace of local P2 at weak and strong coupling and prove closed formulae for the Stokes constants as divisor sum functions and for the perturbative coefficients as values of L-functions. We argue that a full-fledged strong-weak resurgent symmetry is at play, exchanging the perturbative/non-perturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a global net of relations connecting the perturbative series and the discontinuities in the dual regimes, which is built upon the analytic properties of the L-functions with coefficients given by the Stokes constants and the resurgence of the q-series acting as their generating functions. Then, we show that the latter are holomorphic quantum modular forms and are reconstructed by the median resummation of their asymptotic expansions. This leads us to discuss new perspectives on the resurgence of formal power series and quantum modularity and to introduce the notion of modular resurgence. This thesis is based on the publications [1-3]. The author's graduate work also included the publications [4, 5], which are unrelated to the topics presented here.