  Topological superconductors harbor, at their boundaries and vortex cores,
zero-energy Majorana bound states, which can be the building blocks in
fault-tolerant topological quantum computing. Planar Josephson junctions host
such topological superconducting phases, highly tunable by external magnetic
field or phase difference between the superconducting leads. Despite many
theoretical and experimental studies, the signatures of the transition to a
topological superconducting phase, based on minima in the critical supercurrent
$I_c$ flowing across the junction, $0$-$\pi$ transition in the ground state
junction phase and their anisotropic magnetic-field response have remained
unsettled. Using rigorous numerical calculations with several
experimentally-relevant parameter settings, we show that $I_c$ and
$\varphi_{_{\rm GS}}$ cannot indicate unambiguously topological transition in
any realistic planar junctions. Furthermore, the anisotropic variations of
$I_c$ and $\varphi_{_{\rm GS}}$ with in-plane magnetic field appear in
junctions that are undoubtedly in trivial superconducting phase, raising
concerns on the effectiveness of these probes in identifying topological
transitions in planar junctions. We discuss possible strategies to confirm a
topological superconducting phase in these platforms.
