We carry out numerical studies of static packings of frictionless superellipsoidal particles in three spatial dimensions. We consider more than $200$ different particle shapes by varying the three shape parameters that define superellipsoids... We characterize the structural and mechanical properties of both disordered and ordered packings using two packing-generation protocols. We perform athermal quasi-static compression simulations starting from either random, dilute configurations (Protocol 1) or thermalized, dense configurations (protocol $2$), which allows us to tune the orientational order of the packings. In general, we find that the contact numbers at jamming onset for superellipsoid packings are hypostatic, with $z_J < z_{\rm iso}$, where $z_{\rm iso} = 2d_f$ and $d_f = 5$ or $6$ depending on whether the particles are axi-symmetric or not. Over the full range of orientational order, we find that the number of quartic modes of the dynamical matrix for the packings always matches the number of missing contacts relative to the isostatic value. This result suggests that there are no mechanically redundant contacts for ordered, yet hypostatic packings of superellipsoidal particles. Additionally, we find that the packing fraction at jamming onset for disordered packings of superellipsoidal particles can be collapsed using two particle shape parameters, e.g. the asphericity $\mathcal{A}$ and reduced aspect ratio $\beta$ of the particles. (read more)