We're putting aside the confusion about scallions vs. green onions for a second (for a record they're the same thing) to talk about how to store green onions. When you pick up a bunch of green onions from the grocery store, you'll find that they're delicately packed into bundles and stacked into piles. Because of their thin skin, green onions don't last long without proper care. Don't just throw them in the back of your fridge and toss a package of deli meat, more produce, and a bottle of sparkling water on top. Treat them with some care, dang it!
Think of green onions (or scallions) like flowers. They need moisture to stay fresh and are best when they're upright. So we're going to make a bouquet of them: Grab a mason jar or tall glass with water and submerge the green onion's root in an inch or two of water. Leave the green top out of the water, while keeping the white part generously damp. From here, you can store them on a windowsill because who doesn't need a little bit of sunshine, or in the refrigerator on the bottom shelf.
If you've experienced one too many spills using this method with other types of produce (been there, done that) and want to forgo it altogether, there's another trick that will keep green onions fresh for days. Wrap the green onions gently in a damp paper towel, tuck them into an airtight plastic bag, and store them in the crisper drawer of your refrigerator. And always, always, always label and date the bag or container that you're storing scallions in so that you know exactly what they are (no suspicious glances or questions of "are they chives or scallions?") and when to toss them out (if they start to wilt, that's a good indication that they've seen better days).