We present a numerical study of the structure and stability of laminar isothermal flows formed by two counterflowing jets of an incompressible Newtonian fluid. We demonstrate that symmetric counterflowing jets with identical mass flow rates exhibit multiple steady states and, in certain cases, time-dependent (periodic) steady states. Two geometric configurations were studied based on the inlet jet shapes: planar and axisymmetric. Stagnation flows formed by planar counterflowing jets exhibit both steady-state multiplicity and time-dependent behaviour, while axisymmetric jets exhibit only a steady-state multiplicity. A linearized bifurcation and stability analysis based on the continuity and Navier–Stokes equations revealed transitions between a single (symmetric) steady state and multiple steady states or periodic steady states. The dimensionless quantities forming the parameter space of this system are the inlet Reynolds number (R$e$) and a geometric aspect ratio ($\alpha$), equal to the jet inlet characteristic length (used for calculating R$e$) divided by the jet separation. The boundaries separating different flow regimes have been identified in the (R$e$, $\alpha$) parameter space. The resulting flow maps are useful for the design and operation of counterflow jet reactors.