This paper investigates the effects of the track geometry irregularities on the wheel-rail dynamic interactions and the rail fatigue initiation through the application of the Dang Van criterion, that supposes an elastic shakedown of the structure. The irregularities are modelled , using experimental data, as a stochastic field which is representative of the considered railway network. The tracks thus generated are introduced as the input of a railway dynamics software to characterize the stochastic contact patch and the parameters on which it depends: contact forces and wheelset-rail relative position. A variance-based global sensitivity analysis is performed on quantities of interest representative of the dynamic behaviour of the system, with respect to the stochastic geometry irregularities and for different curve radius classes and operating conditions. The estimation of the internal stresses and the fatigue index being more time-consuming than the dynamical simulations, the sensitivity analysis is performed through a metamodel, whose input parameters are the wheel-rail relative position and velocity. The coefficient of variation of the number of fatigue cycles, when the simulations are performed with random geometry irregularities, varies between 0.13 and 0.28. In a large radius curve, the most influent irregularity is the horizontal curvature, while, in a tight curve, the gauge becomes more important.