We demonstrate a class of two-dimensional steady viscous flows which have singular continuous (fractal) Fourier spectra. Such flows represent a novel intermediate stage between order and Lagrangian chaos: the motion of individual fluid particles in them is neither entirely correlated nor completely disordered. We describe qualitatively and numerically the solutions of the two-dimensional Stokes equations for a flow of a viscous incompressible fluid past a periodic array of solid cylinders. The equations are solved on a square with periodic boundary conditions; the central part of the domain is occupied by the solid body with no-slip condition on its border. For the observable associated with the moving fluid particle (e.g. instantaneous velocity of the particle), the temporal autocorrelation function decays; the decay rate is prescribed by the power law. Estimation of the Fourier spectrum indicates that the spectral measure is supported by the multifractal set. Furthermore, this simple steady two-dimensional flow pattern exhibits anomalous transport properties. Existence of this unusual state is caused by the power-like singularities of passage times which develop along the particle paths near the surface of the solid body.