We present a coupled groundwater-stream model of catchment-scale solute transport, using a Lagrangian stochastic advective-sorptive travel time approach. We consider distributed solute input over an entire catchment and investigate the resulting stream solute breakthrough, subject to the possible solute spreading mechanisms: (1) variable groundwater advection and solute mass transfer between mobile and immobile groundwater zones and (2) in-stream advection, mixing, and solute mass transfer between stream water and hyporheic zone. Among these mechanisms we show that fractal solute spreading over a wide timescale range is, for realistic parameter values, obtained in the stream only for the condition combination of both variable solute advection and solute mass transfer in the groundwater, with mean groundwater advection to mass transfer rate ratio that falls within a certain value range and with only a small fraction of solute input mass following fast overland and/or storm soil water flow to the stream.