We determine the local density of states (LDOS) of one-dimensional incommensurate charge density wave (CDW) states in the presence of a strong impurity potential, which is modeled by a boundary. We find that the CDW gets pinned at the impurity, which results in a singularity in the Fourier transform of the LDOS at momentum 2k_F. At energies above the spin gap we observe dispersing features associated with the spin and charge degrees of freedom respectively. In the presence of an impurity magnetic field we observe the formation of a bound state localized at the impurity. All of  our results carry over to the case of 1D Mott insulators by exchanging the  roles of spin and charge degrees of freedom. We discuss the implications of our result for scanning tunneling microscopy experiments on spin-gap systems such as two-leg ladder cuprates.