The development and formulation of printable inks for extrusion-based 3D bioprinting has been a major challenge in the field of biofabrication. Inks, often polymer solutions with the addition of crosslinking to form hydrogels, must not only display adequate mechanical properties for the chosen application but also show high biocompatibility as well as printability. Here we describe a reproducible two-step method for the assessment of the printability of inks for bioprinting, focussing firstly on screening ink formulations to assess fibre formation and the ability to form 3D constructs before presenting a method for the rheological evaluation of inks to characterise the yield point, shear thinning and recovery behaviour. In conjunction, a mathematical model was formulated to provide a theoretical understanding of the pressure-driven, shear thinning extrusion of inks through needles in a bioprinter. The assessment methods were trialled with a commercially available crème, poloxamer 407, alginate-based inks and an alginate-gelatine composite material. Yield stress was investigated by applying a stress ramp to a number of inks, which demonstrated the necessity of high yield for printable materials. The shear thinning behaviour of the inks was then characterised by quantifying the degree of shear thinning and using the mathematical model to predict the window of printer operating parameters in which the materials could be printed. Furthermore, the model predicted high shear conditions and high residence times for cells at the walls of the needle and effects on cytocompatibility at different printing conditions. Finally, the ability of the materials to recover to their original viscosity after extrusion was examined using rotational recovery rheological measurements. Taken together, these assessment techniques revealed significant insights into the requirements for printable inks and shear conditions present during the extrusion process and allow the rapid and reproducible characterisation of a wide variety of inks for bioprinting.