  We study the parabolic Anderson model (PAM) \begin{equation}
  {\partial \over \partial t}u(t,x) =\frac{1}{2}\Delta u(t,x) + u(t,x)\xi(x),
\quad t>0, x\in \mathbb{R}^d, \quad \text{and} \quad
  u(0,x) \equiv 1, \quad \forall x\in \mathbb{R}^d,
  \end{equation} where $\xi$ is spatial white noise on $\mathbb{R}^d$ with $d
\in\{2,3\}$. We show that the peaks of the PAM are macroscopically
multifractal. More precisely, we prove that the spatial peaks of the PAM have
infinitely many distinct values and we compute the macroscopic Hausdorff
dimension (introduced by Barlow and Taylor) of those peaks. As a byproduct, we
obtain the exact spatial asymptotics of the solution of the PAM. We also study
the spatio-temporal peaks of the PAM and show their macroscopic
multifractality. Some of the major tools used in our proof techniques include
paracontrolled calculus and tail probabilities of the largest point in the
spectrum of the Anderson Hamiltonian.
