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We introduce SPIRAL, a SuPerlinearly convergent Incremental pRoximal ALgorithm, for solving nonconvex regularized finite sum problems under a relative smoothness assumption. In the spirit of SVRG and SARAH, each iteration of SPIRAL consists of an inner and an outer loop. It combines incremental and full (proximal) gradient updates with a linesearch. It is shown that when using quasi-Newton directions, superlinear convergence is attained under mild assumptions at the limit points. More importantly, thanks to said linesearch, global convergence is ensured while it is shown that unit stepsize will be eventually always accepted. Simulation results on different convex, nonconvex, and non-Lipschitz differentiable problems show that our algorithm as well as its adaptive variant are competitive to the state of the art.