An integrated circuit ('IC') that includes a configurable routing fabric with controllable storage elements is described. The routing fabric provides a communication pathway that routes signals to and from source and destination components. The routing fabric may provide the ability to selectively store the signals passing through the routing fabric within the storage elements of the routing fabric. In this manner, a source or destination component may continually perform operations (e.g., computational or routing) irrespective of whether a previous signal from or to such a component is stored within the routing fabric. The source and destination components include configurable logic circuits, configurable interconnect circuits, and various other circuits that receive or distribute signals throughout the IC.