We study the dynamics of strings near spacetime singularities. We consider gravitational-wave backgrounds with a singularity of the type {vert bar}{ital U}{vert bar}{sup {minus}{beta}}, {ital U} being a null coordinate. (The case with a {delta}({ital U}) shock-wave singularity turns out to be similar to the {beta}=1 case.) New features in the string behavior appear: when {beta}{ge}2, the string does not propagate through the gravitational wave and it escapes to infinity grazing the singularity plane {ital U}=0; one transverse coordinate does not oscillate in time (neither classically nor quantum mechanically) and the tunnel effect does not take place. The expectation value of the mass squared {l angle}{ital M}{sub {gt}}{sup 2}{r angle} and mode number {l angle}{ital N}{sub {gt}}{r angle} operators and of the energy-momentum tensor are computed. When the transverse size ({rho}{sub 0}) of the gravitational-wave front is infinite, divergences in {l angle}{ital M}{sub {gt}}{sup 2}{r angle} and {l angle}{ital N}{sub {gt}}{r angle} appear for 1{le}{beta}{lt}2 and 3/2{le}{beta}{lt}2, respectively. We argue that the short-distance spacetime singularity at {ital U}=0 is not responsible for these divergences, but the infinite amount of energy carried by the gravitational wave when {rho}{sub 0}={infinity}. In summary, the propagation of strings through these singular spacetimes is provenmore » to be physically meaningful for {beta}{ge}2 and {beta}{lt}1. We conjecture that this is also the case for 1{le}{beta}{lt}2.« less