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Wave scattering from a cylinder with a tensor impedance surface is investigated based on the Lorentz-Mie theory. A practical example of such a cylinder is a subwavelength metallic rod with helical dielectric-filled corrugations. The investigation is performed with the aim to maximize scattering cross-section by tailoring the surface impedance of cylindrical scatterers. For the normally incident TEz and TMz waves the required surface impedance of a subwavelength cylinder can be produced by longitudinal (axial) and transverse (circumferential) corrugations, respectively. It is shown that such corrugations induce superscattering at multiple frequencies, which can be widely tuned with either or both the size and permittivity of dielectric-filled corrugations. In the microwave band, this effect is demonstrated to be robust to material losses and is validated against the full-wave simulations and experiment. For the TEz waves the enhanced scattering from the cylinder is found to have a broad frequency bandwidth, provided that the relative permittivity of corrugations is low or equal unity. In the latter case, the corrugated cylinder acts as an all-metal superscatterer. For such cylinders the near-field measurements are implemented and provide the first experimental evidence of the superscattering phenomenon for all-metal objects. In addition to multifrequency superscattering, the dielectric-filled corrugations are shown to provide multifrequency cloaking of the cylinder under the incidence of the TMz waves. Simultaneous superscattering and cloaking at multiple frequencies distinguishes corrugated cylinders from other known practicable scatterers for potential applications in antenna designing, sensing, and energy harvesting.