In the so-called pasta phases predicted to occur in neutron-star crusts, protons are able to move easily over large distances because the nuclear matter regions are extended in space. Consequently, electrical currents can be carried by protons, an effect not possible in conventional crystalline matter with isolated nuclei. With emphasis on the so-called lasagna phase, which has sheet-like nuclei, we describe the magnetic properties of the pasta phases allowing for proton superconductivity. We predict that these phases will be Type-II superconductors and we calculate the energy per unit length of a flux line, which is shown to be strongly anisotropic. If, as seems likely, the pasta structure is imperfect, flux lines will be pinned and matter will behave as a good electrical conductor and flux decay times will be long. We describe some possible astrophysical manifestations of our results.