We present a first-principles investigation of structural and elastic properties of experimentally observed phases of bulk SrRuO$_3$ - namely orthorhombic, tetragonal, and cubic - by applying density functional theory (DFT) approximations. At first, we focus our attention on the accuracy of calculated lattice constants in order to find out DFT approaches that best represent the crystalline structure of SrRuO$_3$, since many important physical quantities crucially depend on change in volume. Next, we evaluate single-crystal elastic constants, mechanical stability, and macroscopic elastic parameters trying to at least partially compensate for the existing lack of information about these fundamental features of SrRuO$_3$. Finally, we analyze the anomalous behavior of low-temperature orthorhombic phase under $C_{44}$ related shear deformation. It turns out that at critical strain values the system exhibits a distinct deviation from the initial behavior which results in an isosymmetric phase transition. Moreover, under $C_{44}$ related shear deformation tetragonal SrRuO3 becomes mechanically unstable raising an open question of what makes it experimentally observable at high temperatures.
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