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Disordered systems present multifractal properties at criticality. Inparticular, 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 revisitthese ideas to incorporate more recent findings: (i) whenever a multifractalmeasure $w(r)$ normalized over space $ \sum_r w(r)=1$ occurs in a randomsystem, it is crucial to distinguish between the typical values and thedisorder averaged values of the generalized moments $Y_q =\sum_r w^q(r)$, sincethey 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 inducesa lack of self-averaging at critical points for thermodynamic observables, inparticular for the order parameter. After this general discussion valid for anyrandom critical point, we apply these ideas to random polymer models that canbe studied numerically for large sizes and good statistics over the samples. Westudy the bidimensional wetting or the Poland-Scheraga DNA model with loopexponent $c=1.5$ (marginal disorder) and $c=1.75$ (relevant disorder). Finally,we argue that the presence of finite Griffiths ordered clusters at criticalitydetermines 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$f(\alpha)$.

On the multifractal statistics of the local order parameter at random critical points : application to wetting transitions with disorder