  Triple nodal points are degeneracies of energy bands in momentum space at
which three Hamiltonian eigenstates coalesce at a single eigenenergy. For
spinless particles, the stability of a triple nodal point requires two
ingredients: rotation symmetry of order three, four or six; combined with
mirror or space-time-inversion symmetry. However, despite ample studies of
their classification, robust boundary signatures of triple nodal points have
until now remained elusive. In this work, we first show that pairs of triple
nodal points in semimetals and metals can be characterized by Stiefel-Whitney
and Euler monopole invariants, of which the first one is known to facilitate
higher-order topology. Motivated by this observation, we then combine symmetry
indicators for corner charges and for the Stiefel-Whitney invariant in two
dimensions with the classification of triple nodal points for spinless systems
in three dimensions. The result is a complete higher-order bulk-boundary
correspondence, where pairs of triple nodal points are characterized by
fractional jumps of the hinge charge. We present minimal models of the various
species of triple nodal points carrying higher-order topology, and illustrate
the derived correspondence on Sc$_3$AlC which becomes a higher-order
triple-point metal in applied strain. The generalization to spinful systems, in
particular to the WC-type triple-point material class, is briefly outlined.
