  We study the band topology of the superconducting nodal-line semimetal
(SC-NLSM) protected by the inversion symmetry with and without the spin-orbital
coupling. Without the spin-orbital coupling, both the $s$-wave SC-NLSM and the
chiral $p$-wave SC-NLSM are topologically nontrivial and can be described by
the nonzero winding number. Based on the Wilson loop method, we verify that
they are both the fragile topological superconductors, namely, their nontrivial
band topologies can be moved off by coupling to additional topologically
trivial bands. The fragile topological phase persists in spinful system with
the time-reversal symmetry when a spin-orbital coupling term is added. For the
spinful system, both the $p$-wave SC-NLSM and the $s$-wave SC-NLSM are
second-order fragile topological superconductors. We propose that the fragile
topology in the SC-NLSM system depends strongly on the degeneracy of the
Majorana zero modes and the parity of the superconducting gap function.
Interestingly, in presence of a vortex line, the spinful $s$-wave SC-NLSM
system hosts two pairs of stable Majorana zero modes in the vortex core.
