  Let $\xi$ denote space-time white noise, and consider the following
stochastic partial differential equations:
  (i) $\dot{u}=\frac{1}{2} u" + u\xi$, started identically at one; and (ii)
$\dot{Z}=\frac12 Z" + \xi$, started identically at zero.
  It is well known that the solution to (i) is intermittent, whereas the
solution to (ii) is not. And the two equations are known to be in different
universality classes.
  We prove that the tall peaks of both systems are multifractals in a natural
large-scale sense. Some of this work is extended to also establish the
multifractal behavior of the peaks of stochastic PDEs on
$\mathbf{R}_+\times\mathbf{R}^d$ with $d\ge 2$. G. Lawler has asked us if
intermittency is the same as multifractality. The present work gives a negative
answer to this question.
  As a byproduct of our methods, we prove also that the peaks of the Brownian
motion form a large-scale monofractal, whereas the peaks of the
Ornstein--Uhlenbeck process on $\mathbf{R}$ are multifractal.
  Throughout, we make extensive use of the macroscopic fractal theory of M.T.
Barlow and S. J. Taylor (1989, 1992). We expand on aspects of the Barlow-Taylor
theory, as well.
