At 3C or 37F, we could expect precipitation like rain or sleet to be falling. The freezing point of water is 0C or 32F, so it's not quite cold enough for totally frozen precipitation. Whether it is rain or sleet with really depend on the temperature of air at higher altitudes, where the condensation is occurring. It may also depend on the chemical compositions of the water vapor, as highly polluted vapor may be more resistant to crystallization or freezing.
At 0C or 32F and below, we can expect that precipitation will be totally frozen. At this temperature, we are more likely to see snow or hail. Again, the type depends on the temperatures at high altitudes and how quickly water vapor is condensing and freezing. Snow forms and falls relatively quickly compared to hail, which may be carried around by winds at high altitudes gathering more water droplets until it is sufficiently heavy to fall.