  Using the conventional $T$-matrix approach, we discuss gapped phases in 1D,
2D, and 3D spin systems (both with and without a long range magnetic order)
with bond disorder and with weakly interacting bosonic elementary excitations.
This work is motivated by recent experimental and theoretical activity in
spin-liquid-like systems with disorder and in the disordered interacting boson
problem. In particular, we apply our theory to both paramagnetic low-field and
fully polarized high-field phases in dimerized spin-$\frac12$ systems and in
integer-spin magnets with large single-ion easy-plane anisotropy $\cal D$ with
disorder in exchange coupling constants (and/or $\cal D$). The elementary
excitation spectrum and the density of states are calculated in the first order
in defects concentration $c\ll1$. In 2D and 3D systems, the scattering on
defects leads to a finite damping of all propagating excitations in the band
except for states lying near its edges. We demonstrate that the analytical
approach is inapplicable for states near the band edges and our numerical
calculations reveal their localized nature. We find that the damping of
propagating excitations can be much more pronounced in considered systems than
in magnetically ordered gapless magnets with impurities. In 1D systems, the
disorder leads to localization of all states in the band, while those lying far
from the band edges (short-wavelength excitations) can look like conventional
wavepackets.
