1. When you are at the grocery store, look for a bunch that have a good root structure. Don't pick ones that have tiny root stubs as they won't grow very well.
2. When it's time to use it, chop your green onion only down to a short bit above the white part.
3. Use an old yogurt container or a small vase, fill up with some water, and drop the green onion stub in.
4. Place somewhere where there is a bit of natural light. The green onion will continue to grow. When it grows long enough, chop down what you need and repeat. It will keep growing and growing...
If you leave the stubs submerged in water too long, they will begin to disintegrate. What you need to do is every three days or so, pour out the water and add only enough to cover the roots lightly. Leave the water that low for two or three days so the stem material can dry out. Then you can add more/a higher level of fresh water the next time around. Just repeat this process and your green onions will grow and not fall apart.