Mild traumatic brain injuries remains a global public health challenge. The primary injury biomechanics still remains unclear. To model the soft solid (i.e. brain) deformation it is important to characterise the material accurately. A robust mechanical characterisation of brain matter is an essential requirement for correctly estimating the brain deformation in the event of an head impact like in concussion. A correct modelling of nonlinear, dissipative and dispersive behaviour of brain to describe the shear wave propagation in brain is the objective of this work. In this work, we have carefully reviewed the literature and developed frequency-dependent attenuation power laws for 12 different regions of brain varying across 8 different animal species using 212 Prony-series [1]. These power-laws describe the varying attenuating and dispersive behaviour of shear waves in brain. On the other hand, the recently observed formation of shear shock waves in brain which results in very high local acceleration demonstrating the strong nonlinearity in the brain matter [2]. This nonlinear behaviour is carefully studied with these attenuation laws for characterisation. The various attenuation power laws in different regions of brain are shown in Figure 1. Furthermore, the nonlinear parameter (β) responsible for shear shock formation of brain was found to be β = 44.24 ± 8.23. This was done using the experimental data from porcine measurements with an attenuation power law of: 0.06ω1.05, where ω is the angular frequency, and the reference shear speed is 2.10 m/s at 75 Hz. It highlights the importance of modelling correctly the competing effects of nonlinearity and attenuation in brain deformation to obtain more biofidelic estimates.