  High-accuracy gravitational-wave modeling demands going beyond linear,
first-order perturbation theory. Particularly motivated by the need for
second-order perturbative models of extreme-mass-ratio inspirals and black hole
ringdowns, we present practical spherical-harmonic decompositions of the
Einstein equation, Regge-Wheeler-Zerilli equations, and Teukolsky equation at
second perturbative order in a Schwarzschild background. Our formulations are
covariant on the $t$--$r$ plane and on the two-sphere, and we express the field
equations in terms of gauge-invariant metric perturbations. In a companion
Mathematica package, PerturbationEquations, we provide these invariant formulas
as well as the analogous formulas in terms of raw, gauge-dependent metric
perturbations. Our decomposition of the second-order Einstein equation, when
specialized to the Lorenz gauge, was a key ingredient in recent second-order
self-force calculations [Phys. Rev. Lett. 124, 021101 (2020); ibid. 127, 151102
(2021); ibid. 130, 241402 (2023)].
