Transfer operator-based approaches have been successfully applied to the extraction of coherent features in flows. Transfer operators describe the evolution of densities under the action of the flow. They can be efficiently approximated within a set-oriented numerical framework and spectral properties of the resulting stochastic matrices are used to extract finite-time coherent sets. Also finite-time entropy, a density-based stretching quantity similar to finite-time Lyapunov exponents, is conveniently approximated by means of the discretized transfer operator. Transfer operator-based computational methods are purely probabilistic and derivative-free. Therefore, they can also be applied in settings where derivatives of the flow map are hardly accessible. In this paper, we summarize the theory and numerics behind the transfer operator approach and then introduce a straightforward extension, which allows us to study coherent structures in complex flows on triangulated surfaces. We illustrate our general computational framework with the well-known periodically driven double-gyre flow. To demonstrate the applicability of the approach for complex flows, we consider an approximation of the surface ocean flow, obtained by a numerical solution of the incompressible surface Navier-Stokes equation in a complicated geometry on the sphere.