Nonlinear electrodynamics (NLE) is an umbrella term for nonlinear modifications of Maxwell’s theory. The first NLE theories appeared as quantum corrections to Maxwell’s electrodynamics or in order to cure point charge singularities. Our focus is on NLE fields in the context of gravitational theory, their impact on the laws of black hole thermodynamics and the general geometric properties of spacetimes. Using a perturbative approach, we show that the solution for the Schwarzschild black hole placed in an asymptotically homogeneous test magnetic field in the Born–Infeld and Euler–Heisenberg theories receives NLE corrections. We prove that the wellknown results from Maxwell’s theory, such as no-soliton theorems and the absence of null electromagnetic fields in static spacetimes, still hold in the NLE case but may be circumvented by stealth field configurations. In the form of several no-go theorems, we summarise the general obstructions that limit the possibility of black hole regularisation using NLE fields. Our results suggest that physically plausible NLE theories do not give rise to singularity-free solutions. Furthermore, we formulate the sufficient conditions that imply the isometry-compatible block-diagonal form of the metric for theories consisting of solely NLE fields and NLE fields combined with scalar fields. Finally, we revisit the laws of black hole thermodynamics with NLE fields and derive the first law by means of the covariant phase space approach. With a special emphasis on the treatment of NLE Lagrangian parameters, we resolve the tension between the generalised Smarr formula and the first law of black hole thermodynamics in a general case.