Surface-localised states in the form of Fermi arcs emerge at the interfaces between three-dimensional topological semimetals and their surrounding vacuum, serving as primary signatures of momentum-space topology accessible to modern spectroscopic methods. These arcs encode information about topological singularities — nodal points where multiple bands converge — through their geometric properties. Specifically, the arcs terminate tangentially at the edges of Fermi-surface projections, delineating where surface modes transition into bulk extended states. The arcs' multiplicity at each node corresponds directly to the underlying topological charge (Berry-curvature monopole magnitude), thereby offering a window into momentum-space invariants via angle-resolved photoemission experiments. This work establishes a systematic framework for computing surface state configurations applicable to arbitrary nodal degeneracies and band dispersions: (1) nodes with twofold through multifold band multiplicities; and (2) bands exhibiting diverse momentum-dependence patterns, including linear, nonlinear, isotropic, and anisotropic regimes. We additionally examine topologically distinct nodal types where monopole charges vanish but dipolar Berry-curvature distributions appear, clarifying the surface-state landscape in such unconventional scenarios.