We demonstrate a novel dispersion-scan (d-scan) scheme for single-shot temporal characterization of ultrashort laser pulses. The novelty of this method relies on the use of a highly dispersive crystal featuring antiparallel nonlinear domains with a random distribution and size. This crystal, capable of generating a transverse second-harmonic signal, acts simultaneously as the dispersive element and the nonlinear medium of the d-scan device. The resulting in-line architecture makes the technique very simple and robust, allowing the acquisition of single-shot d-scan traces in real time. In addition, the technique can be further simplified by avoiding the need of dispersion pre-compensation. The retrieved pulses are in very good agreement with independent FROG measurements. We also apply the new single-shot d-scan to a TW-class laser equipped with a programmable pulse shaper, obtaining an excellent agreement between the applied and the d-scan retrieved dispersions.