  Norm resolvent approximation for a wide class of point interactions in one
dimension is constructed. To analyse the limit behaviour of Schr\"odinger
operators with localized singular rank-two perturbations coupled with
{\delta}-like potentials as the support of perturbation shrinks to a point, we
show that the set of limit operators is quite rich. Depending on parameters of
the perturbation, the limit operators are described by both the connected and
separated boundary conditions. In particular an approximation for a
four-parametric subfamily of all the connected point interactions is built. We
give examples of the singular perturbed Schr\"odinger operators without
localized gauge fields, which converge to point interactions with the
non-trivial phase parameter. We also construct an approximation for the point
interactions that are described by different types of the separated boundary
conditions such as the Robin-Dirichlet, the Neumann-Neumann or the Robin-Robin
types.
