The linear and nonlinear evolution of two configurations of interest for the solar corona are presented. The first case examined refers to a cylindrically symmetric twisted magnetic flux tube which is intended to model a solar coronal loop which has evolved quasi-statically through sequences of equilibria with increasing twist due to the application of localized photospheric vortex flows. It is well known that when the twist exceeds a critical value that depends on its radial profile and on the loop length, the loop becomes kink-unstable. The nonlinear evolution of the instability is followed using a three-dimensional magnetohydrodynamic (MHD) code in cylindrical geometry in a magnetic field configuration where the net axial current is zero and radially confined, so that the magnetic field is an axial potential one in the external regions. It is shown that, during the nonlinear phase of the instability, loops develop current sheets and consequently their evolution becomes resistive with the occurrence of magnetic reconnection. The details of the reconnection phenomenon and the resistive dissipation mechanism are examined, along with the impact of the results in the understanding of coronal activity. The second case refers to a neutral sheet embedded in a plane fluid wake, which is a simplified representation of a streamer beyond the underlying coronal helmet. This wake- neutral sheet configuration is characterized by two parameters which vary with distance from the Sun: the ratio of the cross-stream velocity scale to the neutral sheet width and the ratio of the typical Alfvén velocity to the typical flow speed far from the neutral sheet. Depending on the values of these parameters, the linear theory predicts that three kinds of instability can develop when this system is perturbed: a tearing instability and two ideal fluid instabilities with different cross-stream symmetries (varicose and sinuous). In the innermost, magnetically-dominated region beyond the helmet cusp, we find that the streamer is resistively and ideally unstable, evolving from tearing-type reconnection in the linear regime to a nonlinear varicose fluid instability. Travelling magnetic islands are formed and the centre of the wake is accelerated and broadened slightly. Past the Alfvén point, where the kinetic energy exceeds the magnetic energy, the tearing mode is suppressed but an ideal sinuous fluid mode can develop, producing additional acceleration up to typical slow wind speeds and substantial broadening of the wake. Farther from the Sun, the system becomes highly turbulent due to the development of ideal secondary instabilities, thus halting the acceleration and producing strong filamentation throughout the core of the wake.