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.
And they're not noise-cancelling headphones, so all the ambient noise like bells and honks and cars and people talking still get heard. This also makes them the safest headphones I've ever owned.
On top of sounding great, it also has a microphone so I can answer calls by simply pressing the button over my right ear. There are plenty of convenient controls on this thing.
And most importantly, it's designed for workouts, so it never falls out and is water resistant. I often forget I'm wearing it.
My wife tells me she bought it online for $120 CAD. If you wear headphones while you bike and/or run, it's worth every penny.