We propose an experimental and theoretical framework for the study of capillary filling at the micro-scale. Our methodology enables us to control the fluid flow regime so that we can characterise properties of Newtonian fluids such as their viscosity. In particular, we study a viscous, non-inertial, non-Washburn regime in which the position of the fluid front increases linearly with time for the whole duration of the experiment. The operating shear-rate range of our apparatus extends over nearly two orders of magnitude. Further, we analyse the advancement of a fluid front within a microcapillary in a system of two immiscible Newtonian liquids. We observe a non-Washburn regime in which the front can accelerate or decelerate depending on the viscosity contrast between the two liquids. We then propose a theoretical model which enables us to study and explain both non-Washburn regimes. Furthermore, our theoretical model allows us to put forward ways to control the emergence of these regimes by means of geometrical parameters of the experimental set-up. Our methodology allows us to design and calibrate a micro-viscosimetre which works at constant pressure.