  Let $\xi$ be a singular Gaussian noise on $\mathbb R^d$ that is either white,
fractional, or with the Riesz covariance kernel; in particular, there exists a
scaling parameter $\omega>0$ such that $c^{\omega/2}\xi(c\cdot)$ is equal in
distribution to $\xi$ for all $c>0$. Let $(\xi_\varepsilon)_{\varepsilon>0}$ be
a sequence of smooth mollifications such that $\xi_\varepsilon\to\xi$ as
$\varepsilon\to0$. We study the asymptotics of the moments of the parabolic
Anderson model (PAM) with noise $\xi_\varepsilon$ as $\varepsilon\to0$, both
for large (i.e., $t\to\infty$) and fixed times $t$. This approach makes it
possible to study the moments of the PAM with regular and singular noises in a
unified fashion, as well as interpolate between the two settings. As
corollaries of our main results, we obtain the following:
  $\textbf{(1)}$ When $\xi$ is subcritical (i.e., $0<\omega<2$), our results
extend the known large-time moment and tail asymptotics for the Stratonovich
PAM with noise $\xi$. Our method of proof clarifies the role of the maximizers
of the variational problems (known as Hartree ground states) that appear in
these moment asymptotics in describing the geometry of intermittency. We take
this opportunity to prove the existence and study the properties of the Hartree
ground state with a fractional kernel, which we believe is of independent
interest.
  $\textbf{(2)}$ When $\xi$ is critical or supercritical (i.e., $\omega=2$ or
$\omega>2$), our results provide a new interpretation of the moment blowup
phenomenon observed in the Stratonovich PAM with noise $\xi$. That is, we
uncover that the latter is related to an intermittency effect that occurs in
the PAM with noise $\xi_\varepsilon$ as $\varepsilon\to0$ for $\textit{fixed
finite times}$ $t>0$.
