We demonstrate experimentally that reflectionless scattering modes (RSMs), a generalized version of coherent perfect absorption, can be functionalized to perform reflectionless programmable signal routing. We achieve versatile programmability both in terms of operating frequencies and routing functionality with negligible reflection upon in-coupling, which avoids unwanted signal-power echoes in radio-frequency or photonic networks. We report in-situ observations of routing functionalities like wavelength demultiplexing, including cases where multi-channel excitation requires adapted coherent input wavefronts. All experiments are performed in the microwave domain based on the same irregularly shaped cavity with strong modal overlap that is massively parametrized by a 304-element programmable metasurface. RSMs in our highly overdamped multi-resonance transport problem are fundamentally intriguing because the simple critical-coupling picture for reflectionless excitation of isolated resonances fails spectacularly. We show in simulation that the distribution of damping rates of scattering singularities broadens under strong absorption so that weakly damped zeros can be tuned toward functionalized RSMs.