I've got quite a bit of experience with this kind of situation, and honestly the thing that changes everything is remembering that the other person is probably asking themselves the same questions. She notices that you notice her, so she already knows you exist. The real trap is overthinking it and turning something mundane into an awkward moment. The key is a gradual approach - not some prepared speech like you're selling her a product.
What a lot of the other answers miss is the function of the space where you run into each other. At the gym or a café you can use the environment as a natural excuse. Like you can ask her advice about the machine she's using, ask if the music's getting to her, or just make a light comment about something happening right now. No need to be original, just honest and low-key. And here's the hack nobody mentions: be ready to accept that the conversation lasts 30 seconds and that's fine. No pressure to drag it out artificially. If it clicks, it'll happen naturally next time. If the person stays closed off well you just let it go without making a big deal and you'll smile at each other like before.
Timing matters too. Don't try to approach someone who's got their headphones in, who's clearly in a rush, or who just finished something tough. Wait for a moment when they look available and relaxed. And after that it's simple: you keep it cool, you act like it's no big deal because it really isn't, and you see how they respond. Zero drama, zero crazy expectations, just a normal human exchange.