  Quantum walks are considered in a one-dimensional random medium characterized
by static or dynamic disorder. Quantum interference for static disorder can
lead to Anderson localization which completely hinders the quantum walk and it
is contrasted with the decoherence effect of dynamic disorder having strength
W, where a quantum to classical crossover at time $t_{c}\propto W^{-2}$
transforms the quantum walk into an ordinary random walk with diffusive
spreading. We demonstrate these localization and decoherence phenomena in
quantum carpets of the observed time evolution and examine in detail a dimer
lattice which corresponds to a single qubit subject to randomness.
