  Gravitational lensing offers a powerful tool for exploring the matter
distribution in the Universe. Thanks to their low frequencies and phase
coherence, gravitational waves (GWs) allow for the observation of novel
wave-optics features (WOFs) in lensing, inaccessible to electromagnetic
signals. Combined with the existing accurate source models, lensed GWs can be
used to infer the properties of gravitational lenses. The prospect is
particularly compelling for space-borne detectors, where the high
signal-to-noise ratio expected from massive black hole binary mergers allows
WOFs to be distinguished deep into the weak lensing regime, drastically
increasing the detection probability. Here, we investigate in detail the
capacity of the LISA mission to detect WOFs caused by dark matter halos,
galaxies and the supermassive black holes (SMBHs) within them. We estimate the
total optical depth to be $\lambda_{\rm tot} \sim 6 \times 10 ^{-3}$ for the
loudest binaries of total mass $M_{\rm BBH} \sim 10^6 M_{\odot}$, with the
dominant contribution coming from SMBHs. We also find that WOFs in low-mass
binaries $M_{\rm BBH} \sim 10^4 M_{\odot}$ are more likely due to the central
galaxies. Within our model of gravitational lenses, we predict
$\mathcal{O}(0.1)-\mathcal{O}(1)$ weakly-lensed events to be detectable during
the 5 years of LISA mission, depending on the source population models. We show
that WOFs signatures are very sensitive to the properties of dark-matter halos
with $M_{\rm vir}\in (10^6-10^8)M_\odot$: increasing the compactness parameter
by $\sim 3$ in that range raises the detection rate by $\sim 26$. Additionally,
we show that collective effects from the complex inner halo structure can
further enhance detectability. This suggests that lensed GWs in LISA will be an
excellent probe of dark-matter theories, baryonic and halo sub-structures.
