The level curvature distribution function is studied both analytically and numerically for the case of T-breaking perturbations over the orthogonal ensemble. The leading correction to the shape of the curvature distribution beyond the random matrix theory is calculated using the nonlinear supersymmetric s model and compared to numerical simulations on the Anderson model. It is predicted analytically and confirmed numerically that the sign of the correction is different for T-breaking perturbations caused by a constant vectorpotential equivalent to a phase twist in the boundary conditions, and those caused by a random magnetic field. In the former case it is shown using a nonperturbative approach that quasilocalized states in weakly disordered systems can cause the curvature distribution to be nonanalytic. In two-dimensional (2D) systems the distribution function P(K) has a branching point at K=0 that is related to the multifractality of the wave functions and thus should be a generic feature of all critical eigenstates. A relationship between the branching power and the multifractality exponent d2 is suggested. Evidence of the branch-cut singularity is found in numerical simulations in 2D systems and at the Anderson transition point in 3D systems.