  Disordered systems present multifractal properties at criticality. In
particular, as discovered by Ludwig (A.W.W. Ludwig, Nucl. Phys. B 330, 639
(1990)) on the case of diluted two-dimensional Potts model, the moments
$\bar{\rho^q(r)}$ of the local order parameter $\rho(r)$ scale with a set
$x(q)$ of non-trivial exponents $x(q) \neq q x(1)$. In this paper, we revisit
these ideas to incorporate more recent findings: (i) whenever a multifractal
measure $w(r)$ normalized over space $ \sum_r w(r)=1$ occurs in a random
system, it is crucial to distinguish between the typical values and the
disorder averaged values of the generalized moments $Y_q =\sum_r w^q(r)$, since
they may scale with different generalized dimensions $D(q)$ and $\tilde D(q)$
(ii) as discovered by Wiseman and Domany (S. Wiseman and E. Domany, Phys Rev E
{\bf 52}, 3469 (1995)), the presence of an infinite correlation length induces
a lack of self-averaging at critical points for thermodynamic observables, in
particular for the order parameter. After this general discussion valid for any
random critical point, we apply these ideas to random polymer models that can
be studied numerically for large sizes and good statistics over the samples. We
study the bidimensional wetting or the Poland-Scheraga DNA model with loop
exponent $c=1.5$ (marginal disorder) and $c=1.75$ (relevant disorder). Finally,
we argue that the presence of finite Griffiths ordered clusters at criticality
determines the asymptotic value $x(q \to \infty) =d$ and the minimal value $
\alpha_{min}=D(q \to \infty)=d-x(1) $ of the typical multifractal spectrum
$f(\alpha)$.
