  Centrosymmetric GdRu$_2$Si$_2$ exhibits a variety of multi-Q magnetic states
as a function of temperature and applied magnetic field, including a square
skyrmion-lattice phase. The material's behavior is strongly dependent on the
direction of the applied field, with different phase diagrams resulting for
fields applied parallel or perpendicular to the crystallographic $c$ axis.
Here, we present the results of muon-spin relaxation ($\mu^+$SR) measurements
on single crystals of GdRu$_2$Si$_2$. Our analysis is based on the computation
of muon stopping sites and consideration of zero-point motion effects, allowing
direct comparison with the underlying spin textures in the material. Using
transverse-field $\mu^+$SR with fields applied along either the [001] or [100]
crystallographic directions, we distinguish between the magnetic phases in this
system via their distinct muon response, providing additional evidence for the
skyrmion and meron-lattice phases, while also suggesting the existence of
RKKY-driven muon hyperfine coupling. Zero-field $\mu^+$SR provides clear
evidence for a transition between two distinct magnetically-ordered phases at
39 K.
