The supplied ear bud (headphones) that come with an iPhone or iPod purchase seem to be very cheaply made and do not hold up. Can anybody recommend a brand or two of ear buds that work great with the iPhone and iPod and are much sturdier? Thanks!

All earbuds and headphones with the standard size jack will work with the little MP3 players. Most portable music devices all the way back to cassette Walkmans have the same sized jack. I even use one of the cd tape adapters to play mine through the tape player in my car. It works great and sounds better than the old tapes.

I think the best quality earbuds are made by Nike. They sell them at best buy and they are made specifically for sports use so they tend to stay put.
Skullcandy also has a really good reputation and they come in lots of stylish designs if that's important to you.