  We demonstrate the unique capabilities of the Wigner function, particularly
in its positive and negative parts, for exploring the phase diagram of the
spin$-(\frac{1}{2\!}-\!\frac{1}{2})$ and spin$-(\frac{1}{2}\!-\!1)$
Ising-Heisenberg chains. We highlight the advantages and limitations of the
phase space approach in comparison with the entanglement concurrence in
detecting phase boundaries. We establish that the equal angle slice
approximation in the phase space is an effective method for capturing the
essential features of the phase diagram, but falls short in accurately
assessing the negativity of the Wigner function for the homogeneous
spin$-(\frac{1}{2}\!-\!\frac{1}{2})$ Ising-Heisenberg chain. In contrast, we
find for the inhomogeneous spin$-(\frac{1}{2}\!-\!1)$ chain that an integral
over the entire phase space is necessary to accurately capture the phase
diagram of the system. This distinction underscores the sensitivity of phase
space methods to the homogeneity of the quantum system under consideration.
