Abstract : [Background] Symmetry restoration and configuration mixing in the spirit of the generator coordinate method based on energy density functionals have become widely used techniques in low-energy nuclear structure physics. Recently, it has been pointed out that these techniques are ill-defined for standard Skyrme functionals, and a regularization procedure has been proposed to remove the resulting spuriosities from such calculations. This procedure imposes an integer power of the density for the density dependent terms of the functional. At present, only dated parameterizations of the Skyrme interaction fulfill this condition. [Purpose] To construct a set of parameterizations of the Skyrme energy density functional for multi-reference energy density functional calculations with regularization using the state-of-the-art fitting protocols. [Method] The parameterizations were adjusted to reproduce ground state properties of a selected set of doubly magic nuclei and properties of nuclear matter. Subsequently, these parameter sets were validated against properties of spherical and deformed nuclei. [Results] Our parameter sets successfully reproduce the experimental binding energies and charge radii for a wide range of singly-magic nuclei. Compared to the widely used SLy5 and to the SIII parameterization that has integer powers of the density, a significant improvement of the reproduction of the data is observed. Similarly, a good description of the deformation properties at $A\sim 80$ was obtained. [Conclusions] We have constructed new Skyrme parameterizations with integer powers of the density and validated them against a broad set of experimental data for spherical and deformed nuclei. These parameterizations are tailor-made for regularized multi-reference energy density functional calculations and can be used to study correlations beyond the mean-field in atomic nuclei.