We study the two-dimensional (2D) stochastic Navier-Stokes (SNS) equations in the inertial limit of weak forcing and dissipation. The stationary measure is concentrated close to steady solutions of the 2D Euler equations. For such inertial flows, we prove that bifurcations in the flow topology occur either by changing the domain shape, the nonlinearity of the vorticity-stream-function relation, or the energy. Associated with this, we observe bistable behavior in SNS with random changes from dipoles to unidirectional flows. The theoretical explanation being very general, we infer the existence of similar phenomena in experiments and in some regimes of geophysical flows.
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