Here’s a short, informative piece on the topic:

So, if someone asks for the month of spring, there isn’t just one. Spring is a three-month season. But if you had to pick a single month that feels most like spring in many temperate climates—mild weather, blooming flowers, and longer daylight— (in the north) or October (in the south) often takes the crown.

The answer depends largely on where you live and which definition of "spring" you use—astronomical or meteorological.

In short: March to May (north), September to November (south).