ABSTRACT Stars form in highly magnetized, supersonic turbulent molecular clouds. Many of the tools and models that we use to carry out star formation studies rely upon the assumption of cloud isotropy. However, structures like high-density filaments in the presence of magnetic fields and magnetosonic striations introduce anisotropies into the cloud. In this study, we use the two-dimensional power spectrum to perform a systematic analysis of the anisotropies in the column density for a range of Alfvén Mach numbers ($\operatorname{\mathcal {M}_{\text{A}}}=0.1{\!-\!10}$) and turbulent Mach numbers ($\operatorname{\mathcal {M}}=2{\!-\!20}$), with 20 high-resolution, three-dimensional turbulent magnetohydrodynamic simulations. We find that for cases with a strong magnetic guide field, corresponding to $\operatorname{\mathcal {M}_{\text{A}}}\lt 1$, and $\operatorname{\mathcal {M}}\lesssim 4$, the anisotropy in the column density is dominated by thin striations aligned with the magnetic field, while for $\operatorname{\mathcal {M}}\gtrsim 4$ the anisotropy is significantly changed by high-density filaments that form perpendicular to the magnetic guide field. Indeed, the strength of the magnetic field controls the degree of anisotropy and whether or not any anisotropy is present, but it is the turbulent motions controlled by $\operatorname{\mathcal {M}}$ that determine which kind of anisotropy dominates the morphology of a cloud.