The spin Hall effect (SHE) and its inverse have been widely used to generate and detect spin currents. To date, most experiments focus only on characterizing electrons near the Fermi surface, whereas the SHE, which originates from the spin-orbit interaction, is expected to be energy dependent. Here, we report a tunnelling spectroscopy technique developed to measure the SHE under finite bias voltages. We studied the SHE for typical 5d transition metals. At zero d.c. bias, the obtained spin Hall angles confirm the results from spin-torque experiments. At high bias, the transverse spin Hall signals of these materials exhibit very different voltage dependences. The SHE tunnelling spectra have important implications in pinpointing the mechanisms of the SHE and provide guidelines for engineering high-SHE materials. Moreover, SHE tunnelling spectroscopy can be directly applied to two-dimensional surface states with strong spin-orbit coupling, such as Dirac electrons in topological insulators. © 2014 Macmillan Publishers Limited. All rights reserved.