  We present a detailed study of the magnetic properties of weakly
ferromagnetic/quantum critical Nb$_{1-y}$Fe$_{2+y}$ using muon spin rotation
and relaxation ($\mu$SR). By means of an angular dependent study of the muon
spin rotation signal in applied magnetic fields on a single crystal in the
paramagnetic state we establish the muon stopping site in the crystallographic
lattice of NbFe$_2$. With this knowledge we develop models to describe the muon
spin rotation and relaxation signals in the weakly ferromagnetic, spin density
wave and quantum critical phases of Nb$_{1-y}$Fe$_{2+y}$ and fit the
corresponding experimental data. In particular, we quantify the $\mu$SR
response for quantum critical behavior in Nb$_{1.0117}$Fe$_{1.9883}$ and
extract the influence of residual weak structural disorder. From our analysis,
Nb$_{1-y}$Fe$_{2+y}$ emerges to be uniquely suited to study quantum criticality
close to weak itinerant ferromagnetic order.
