When is a sticks shadow shortest on a sundial?

On a daily basis, Noon, or the solar equivalent of mid-day.

On an annual basis it would depend on where you are.
Since the earth tilts to a maximum of 23.44°, then if one above the latitude 23.44°, or below the latitude -23.44°, then it would be shortest at noon on the summer solstice (June in the northern hemisphere, December in the Southern Hemisphere).

Between the latitudes -23.44° and 23.44°, there would be 2 days a year where the sun is directly overhead.