  The electron spectral shapes corresponding to the low-$Q$ $\beta^-$-decay
transitions $^{151}$Sm$(5/2^-_{\rm g.s.})\to\,^{151}\textrm{Eu}(5/2^+_{\rm
g.s.})$, $^{151}$Sm$(5/2^-_{\rm g.s.})\to\,^{151}\textrm{Eu}(7/2^+_{1})$,
$^{171}$Tm$(1/2^+_{\rm g.s.})\to\,^{171}\textrm{Yb}(1/2^-_{\rm g.s.})$,
$^{171}$Tm$(1/2^+_{\rm g.s.})\to\,^{171}\textrm{Yb}(3/2^-_{1})$,
$^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(1^-_{\rm g.s.})$, and
$^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(0^-_{1})$ have been
computed using beta-decay theory with several refinements for these
first-forbidden nonunique (ff-nu) $\beta^-$ transitions. These ff-nu $\beta^-$
transitions have non-trivial electron spectral shapes with transition nuclear
matrix elements (NMEs) computed by using the microscopic Interacting
Boson-Fermion Model (IBFM-2) for the decays of $^{151}$Sm and $^{171}$Tm, and
the nuclear shell model (NSM) for the decay of $^{210}$Pb. Within the
respective $Q$ windows, the computed ff-nu electron spectral shapes deviate
maximally at sub-percent level from the universal allowed shape, except for the
transition $^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(1^-_{\rm
g.s.})$, where the maximal deviation is some 2.7$\%$. This confirms that the
so-called $\xi$ approximation is fairly good for most of these low-$Q$
$\beta^-$ transitions and thus the allowed shape is a rather good first
approximation. Our computed spectral shapes could be of interest for
experiments aiming to measure the cosmic neutrino background (C$\nu$B), like
the PTOLEMY experiment. We have also derived C$\nu$B cross sections for the
ground-state transitions of the considered nuclei at the $\beta$ endpoint. Our
findings indicate that more work on the atomic mismatch correction is needed in
the future in order to extract reliable and precise C$\nu$B cross sections for
any nuclear target.
