The tone of the story, along with what the power is in itself, can impact how you might want the discovery process to work. For example: let's say someone suddenly has super strength. If you're going for a light-hearted tone, then they could accidentally rip off the door to their house one day while trying to open it. Day-to-day life would suddenly become awkward, as they try to adjust to it. Maybe they want to hide it, or maybe half the village sees it happen. The village, if/when they find out, can react in a lot of different ways too.
If you're going for a more serious or intense approach, then as Mizal and Steve mentioned, emotional life and death situations are a good way bring out. It could be helping someone else, defending yourself, or even getting enraged at something happening. Also think about what the power is and try to be creative in how it newly manifests itself. Make sure it feels accidental, and not too conveniently doing the optimal thing the power could be used for right from the get-go.