  Quasi-one-dimensional systems demonstrate Van Hove singularities in the
density of states $\nu_F$ and the resistivity $\rho$, occurring when the Fermi
level $E$ crosses a bottom $E_N$ of some subband of transverse quantization. We
demonstrate that the character of smearing of the singularities crucially
depends on the concentration of impurities. There is a crossover concentration
$n_c\propto |\lambda|$, $\lambda\ll 1$ being the dimensionless amplitude of
scattering. For $n\gg n_c$ the singularities are simply rounded at
$\varepsilon\equiv E-E_N\sim \tau^{-1}$ -- the Born scattering rate. For $n\ll
n_c$ the non-Born effects in scattering become essential despite $\lambda\ll
1$. The peak of the resistivity is asymmetrically split in a Fano-resonance
manner (however with a more complex structure). Namely, for $\varepsilon>0$
there is a broad maximum at $\varepsilon\propto \lambda^2$ while for
$\varepsilon<0$ there is a deep minimum at $|\varepsilon|\propto n^2\ll
\lambda^2$. The behaviour of $\rho$ below the minimum depends on the sign of
$\lambda$. In case of repulsion $\rho$ monotonically grows with $|\varepsilon|$
and saturates for $|\varepsilon|\gg \lambda^2$. In case of attraction $\rho$
has sharp maximum at $|\varepsilon|\propto \lambda^2$. The latter feature is
due to resonant scattering by quasistationary bound states that inevitably
arise just below the bottom of each subband for any attracting impurity.
