I have these ones below. They're about $20 at most, and they stay in place. I sweat more than any other person I've ever met, and I've never had any problems with performance when I sweat. I've tried some in-ear headphones, but the sweat can cause them to kind of crackle when you move because the "seal" in your ear canal gets all wet and what not. It's kind of hard to explain. I also have some old school ghetto ones that I wear when I wear a beanie while running, and they just go over the top of my head.

When cycling, I usually just use my phone earpiece (in my right ear, of course). It's not comfortable, and the wind is usually too loud to hear anything anyways, but for some reason I still use it.

My headphones that I use when not exercising are Shure E3C (significantly more than $20). They're noise isolating in-ear headphones, but they get really sweaty when I wear them exercising, and pretty much fall out. I wouldn't suggest them, or any like them.

Also, FWIW, I don't think you should have to worry about sweat damaging most earphones/headphones. It might make them not fit well or whatever, but it usually won't damage them at all. Also, I would look for some that only have a cord on one side (like the ones below). That makes them infinitely more comfortable to exercise in.

I use the Philips version of the same style headphone as above and also have had great luck with them.

+1 on headphones during biking. I used to, and it's almost ended badly for me many times so i stopped.

-Alan

If you don't mind the 70s retro look, a good sweatband will intercept the sweat before it can get down and interfere with your headphones. My new favorite is the UA visor. Its pretty amazing at wicking sweat away from face.

+1 on riding without headphones outside. I used to, then realized how stupid it was with all the trafffic in the area and havent done it since then. For running I use the Sennheiser PMX70 sport and have never been happier. They are sweat and water resistant and stick to your head like glue when you put them on. I had trouble with earbuds, even with the little ear wrap thing, falling out all the time and it was a pain, so I upgraded and now i wouldnt even know they were there without the music.

http://www.sennheiserusa.com/newsite/productdetail.asp?transid=500504

They are a little on the spendy side, but I believe in buying good stuff the first time.

Yeah, I have the Sony ones Chunky uses as well. I also have the same problem with earbuds getting all sweaty, and then everything starts to sound like the ocean, and there's just a gross puddle of sweat and earwax....

tritoride28's Sennheiser phones look pretty sweet. Do they go into your ear, or do those little things just sit right outside of your ear? Mostly, I just like the green color.

I do not wear headphones when I am running or cycling outdoors. First of all, I find that there are so many interesting things to discover when I am running, and I feel like I am missing out on them if I let myself get too "in the zone". I know that some people don't mind not hearing traffic when they are running, but it just scares me personally. I like that I can hear when I car is coming behind me, even when I am running opposing traffic. Knowing everything that is going on around you at all times can only help your safety.

Well, according to my GRE score, I don't. Freaking quantum physics messing everything up.