  We demonstrate that quantum-critical spin dynamics can be probed in high
magnetic fields using muon-spin relaxation ($\mu^{+}$SR). Our model system is
the strong-leg spin ladder bis(2,3-dimethylpyridinium) tetrabromocuprate
(DIMPY). In the gapless Tomonaga-Luttinger liquid phase we observe
finite-temperature scaling of the $\mu^{+}$SR $1/T_1$ relaxation rate which
allows us to determine the Luttinger parameter $K$. We discuss the benefits and
limitations of local probes compared with inelastic neutron scattering.
