There are usually around two shooting stars per hour, but the best time to see them is during a meteor shower. This is because there will be many more than usual all over the sky. Shooting stars do occur in the daytime, but can be seen more clearly in the sky at night.
In nearly all showers, the radiant is highest just before dawn, but any time between midnight and dawn gives you a view of most meteors head-on for a more frequent display.