  The sample-function regularity of the random-field solution to a stochastic
partial differential equation (SPDE) depends naturally on the roughness of the
external noise, as well as on the properties of the underlying
integro-differential operator that is used to define the equation. In this
paper, we consider parabolic and hyperbolic SPDEs on
$0,\infty)\times\mathbb{R}^d$ of the form $\partial_t u = L u + g(u) + \dot{F}
\qquad\text{and}\qquad \partial^2_t u = L u + c + \dot{F}, $ with suitable
initial data, forced with a space-time homogeneous Gaussian noise $\dot{F}$
that is white in its time variable and correlated in its space variable, and
driven by the generator $L$ of a genuinely $d$-dimensional L\'evy process $X$.
We find optimal H\"older conditions for the respective random-field solutions
to these SPDEs. Our conditions are stated in terms of indices that describe
thresholds on the integrability of some functionals of the characteristic
exponent of the process $X$ with respect to the spectral measure of the spatial
covariance of $\dot F$. Those indices are suggested by references [45, 46] on
the particular case that $L$ is the Laplace operator on $\mathbb{R}^d$.
