  Stationary probability distributions of one-dimensional random walks on
lattices with aperiodic disorder are investigated. The pattern of the
distribution is closely related to the diffusional behavior, which depends on
the wandering exponent $\Omega$ of the background aperiodic sequence: If
$\Omega<0$, the diffusion is normal and the distribution is extended. If
$\Omega>0$, the diffusion is ultraslow and the distribution is localized. If
$\Omega=0$, the diffusion is anomalous and the distribution is singular, which
shows its complex and hierarchical structure. Multifractal analysis are
performed in order to characterize these distributions. Extended, localized,
and singular distributions are clearly distinguished only by the finite-size
scaling behavior of $\alpha_{\rm min}$ and $f(\alpha_{\rm min})$. The
multifractal spectrum of the singular distribution agrees well with that of a
simple partitioning process.
