You shouldn't wear headphones while you bike. At least that's what I tell my kids. The fact is, when I'm biking alone, I always wear headphones.
I used to wear the buds that come with your smartphone. Then, for my recent birthday in June, my wife surprised me with a pair of Plantronics BackBeat FIT wireless stereo headphones. They're incredible.
Firstly, they're wireless. This is awfully convenient. They connect to my phone via Bluetooth and I can bike a full week before needing to charge them. And charging them is as simple as plugging them into my laptop's USB port for a couple of hours.