  We present the results of muon-spin spectroscopy ($\mu^{+}$SR) measurements
on the molecular spin ladder system (Hpip)$_{2}$CuBr$_{4(1-x)}$Cl$_{4x}$,
[Hpip=(C$_{5}$H$_{12}$N)]. Using transverse field $\mu^{+}$SR we are able to
identify characteristic behaviour in each of the regions of the phase diagram
of the $x=0$ strong-rung spin ladder system (Hpip)$_{2}$CuBr$_4$. Comparison of
our results to those of the dimer-based molecular magnet
Cu(pyz)(gly)(ClO$_{4}$) shows several common features. We locate the crossovers
in partially disordered (Hpip)$_{2}$CuBr$_{4(1-x)}$Cl$_{4x}$ ($x=0.05$), where
a region of behaviour intermediate between quantum disordered and Luttinger
liquid-like is identified. Our interpretation of the results incorporates an
analysis of the probable muon stopping states in (Hpip)$_{2}$CuBr$_4$ based on
density functional calculations and suggests how the muon plus its local
distortion can lead to a local probe unit with good sensitivity to the magnetic
state. Using longitudinal field $\mu^{+}$SR we compare the dynamic response of
the $x=1$ strong-rung material (Hpip)$_{2}$CuCl$_{4}$ to that of the strong-leg
material (C$_{7}$H$_{10}$N)$_{2}$CuBr$_{4}$ (known as DIMPY) and demonstrate
that our results are in agreement with predictions based on interacting
fermionic quasiparticle excitations in these materials.
