We discuss a possibility for simultaneous measurement of different viscometric properties by the oscillating-cup technique. The emphasis is on the following problems related to liquid metal viscometry: identification of non-Newtonian fluid flow and combined viscosity/density measurement. The cases of Bingham and Newtonian fluids are analysed in detail. The basic features of nonlinear fluid flow, viscometer oscillations, modelling of experiment and data processing are systematized and interpreted in terms of linear viscous fluids.