In defect-free, topologically nontrivial materials the concept of topology is formally closely related to the electronic band structure. In this project, we investigate nanoscale samples with discrete spectra, which are made from such materials. We ask how topology enters physical observables in the absence of a band structure. The question is of fundamental interest in itself, but also practical considerations motivate us related, e.g., to a future miniaturization of electronic devices operating with topological materials. Specifically, employing computational tools we search for anomalies, e.g., the topological surface currents in the electrical and magnetic responses of nanorods, 2D-flakes and 3D-grains made of topological matter.