Having in mind singular solutions of smooth reformulations of complementarity problems,
arising unavoidably when the solution in question violates strict complementarity, we study
the behavior of Newton-type methods near singular solutions of nonlinear equations, assuming
that the operator of the equation possesses a strongly semismooth derivative, but is not
necessarily twice differentiable. These smoothness restrictions give rise to peculiarities of the
analysis and results on local linear convergence and asymptotic acceptance of the full step,
the issues addressed in this work. Moreover, we consider not only the basic Newton method,
but also some stabilized versions of it indended for tackling singular (including nonisolated)
solutions. Applications to nonlinear complementarity problems are also dealt with.