  Monolayer WTe$_2$, a centrosymmetric transition metal dichacogenide, has
recently been established as a quantum spin Hall insulator and found
superconducting upon gating. Here we study the pairing symmetry and topological
nature of superconducting WTe$_2$ with a microscopic model at mean-field level.
Surprisingly, we find that the spin-triplet phases in our phase diagram all
host Majorana modes localized on two opposite corners. Even when the
conventional pairing is favored, we find that an intermediate in-plane magnetic
field exceeding the Pauli limit stabilizes an unconventional equal-spin pairing
aligning with the field, which also hosts Majorana corner modes. Motivated by
our findings, we obtain a recipe for two-dimensional superconductors featuring
"higher-order topology" from the boundary perspective: Generally a
superconducting inversion-symmetric quantum spin Hall material whose
normal-state Fermi surface is away from high-symmetry points, such as gated
monolayer WTe$_2$, hosts Majorana corner modes if the superconductivity is
parity-odd. We further point out that this higher-order phase is an
inversion-protected topological crystalline superconductor and study the
bulk-boundary correspondence. Finally, we discuss possible experiments for
probing the Majorana corner modes. Our findings suggest superconducting
monolayer WTe$_2$ is a playground for higher-order topological
superconductivity, and possibly the first material realization for
inversion-protected Majorana corner modes without utilizing proximity effect.
