They are by far always way more trouble than any benefit they give.

These are the ones that you place directly outside your ear, separately, with fully enveloping it.

They are what you will find at most stores now.

I have been seeing the headphones with the actual headband become less and less common.

The ones without the headband just fall out too much.  It can be highly distracting and bothersome to have to keep putting and readjusting those little ear pieces back in your ear ever couple minutes with you are running or jogging.  You want to put them in once and leave them there.

I cannot tell you how many times I have been on the treadmill at the gym or hiking up Flattop and I have had those earbuds fall out of my ear.  Its not worth it at all.  The apple earbuds and Nike earbuds were especially bad and I could never get them to stay in at all!

So what type of headphones do I look for when I am running?

Several things make any headphones you are doing physical activity with good.  I like the wrap around headphones the best.  I think that sometimes they can get a little uncomfortable but they stay on.  I like them to be a lightweight but not too lightweight because I have found that they will flop back and forth if they are.

You also want some that will cancel out the surrounding noise as much as possible.  You do not want to hear a lot of other music around you neither the sounds of all the treadmill fitness equipment at the gym.  Its distracting.

I really would not spend anymore than $20 on any pair of sport headphones.  Its not worth it.  I have bought headphones that were $40 and $50 and I either lost them, had them stolen, or broke them.

Headphones are very fragile and the life expectancy of them is low.   You can rip them, sit on them, break them, and its very easy to do all those things.  Go with a good pair and be ready to replace them every couple months.