•A new variational fully implicit combined field scheme for fluid–body interactions.•Square cylinder with sharp corners initiates lateral edge separation at 95<Re<100.•Rounded square cylinders undergo only vortex-induced motion at low Re.•Motion of the basic square is vortex-induced at low Re and galloping at high Re.•Removal of the corners drastically alters the flow and vibration characteristics. We present a fully implicit combined field scheme based on Petrov–Galerkin formulation for fluid–body interaction problems. The motion of the fluid domain is accounted by an arbitrary Lagrangian–Eulerian (ALE) strategy. The combined field scheme is more efficient than conventional monolithic schemes as it decouples the computation of ALE mesh position from the fluid–body variables. The effect of corner rounding is studied in two-dimensions for stationary as well as freely vibrating square cylinders. The cylinder shapes considered are: square with sharp corners, circle and four intermediate rounded squares generated by varying a single rounding parameter. Rounding of the corners delays the primary separation originating from the cylinder base. The secondary separation, seen solely for the basic square along its lateral edges, initiates at a Reynolds number, Re between 95 and 100. Imposition of blockage lowers the critical Re marking the onset of secondary separation. For free vibrations without damping, Re range is 100–200 and mass ratio, m∗ of each cylinder is 10. The rounded cylinders undergo vortex-induced motion alone whereas motion of the basic square is vortex-induced at low Re and galloping at high Re. The flow is periodic for vortex-induced motion and quasi-periodic for galloping. The lower branch and desynchronization characterize the response of rounded cylinders. For the square cylinder, the components of response are the lower branch, desynchronization and galloping. Removal of the sharp corners of square cylinder drastically alters the flow and vibration characteristics.