The lifting of a deep layer of moisture has the potential to produce heavy rain. In the US, this can occur from a deep layer of Gulf moisture banking up against a slow moving trigger mechanism. Soils can be saturated from rain in a previous storm system or a prolonged rain event from one storm system. Snow melt adds to the amount of runoff. Heavy rain combined with rapid snowmelt is conducive to flash flooding.
Frozen soil acts like saturated soil.
All rainfall will immediately runoff. The pore spaces in very dry soil do not allow water to penetrate at a quick rate. Very heavy rain on bone dry soil can caused flooding. This occurs with heavy convective thunderstorms over arid and semi-arid regions. This can produce local or widespread heavy rain.