Hell, I just toss it in the crisper without even wrapping it.
Plastic container, usually sliced.
If I’m saving it, it’s probably for sandwiches, and I expect to pull out a slice at a time and slap it on whatever sandwiches I make over the next few days. For cooking, I round onion quantities up to the next whole onion, so I never have diced onion left over.
In my world, if you can use half an onion, you can use a whole onion.
But then again, I don’t eat raw onions.
I usually use half an onion at a time. I mold a small piece of aluminum foil against the cut surface(s) of the unused portion and place it in a reusable plastic container (like the ones that lunch meat sometimes is sold in). This keeps fine for a week. Anything other than the foil lets the onion smell permeate the refrigerator. A closed glass container works too, on the rare occasions I have one big enough.
I, too, would be dead by now if using a partial onion invited unbridled bacterial growth. Not something I worry about.
I’m another who usually uses a whole onion even if just a half is called for. But if I’m only using half and I know I’m going to use diced onion within a day, I’ll dice the whole thing and store in a tightly-sealed glass container. E.g., diced onions for the dinner pizza, remainder used in the breakfast omelets. If I’m not sure when I’ll use the rest, I cut it in half and store the root half in the fridge, also in a sealed glass container. I almost always use them up within the week and they’re fine, with little compromise in quality.
I have a dedicated onion bowl that has never contained anything else. It’s an ugly little yellow Tupperware® container that has never let me down.
Sometimes diced, sometimes chunks. I always get a week’s storage.
mmm
I don’t know if cutting the root end last keeps it fresher longer, but I always start from the sprout end. The reason is that the root end holds the whole structure together, making it easier to work with. I just wrap it in plastic wrap and place it wherever it will fit in the fridge. I keep my butter stored in the freezer and take out only one stick at a time, to protect it from onionosity in the fridge, because butter seems more susceptible to the influence than anything else. Yellow onions get cooked, and for raw onion applications I use red ones. The red onions available around here tend to be huge, so I always have some red to save; it seems to have drier flesh and to keep a relatively long time.
I use one of these, took. It also keeps the onion smell from getting out to be picked up by food in the fridge.
Refrigerated onions aren’t quite as strong as fresh ones, but they’ll do in a recipe (like onion rings) that doesn’t require a crisp onion.
Whenever possible, I usually use a whole onion. But when I don’t, I keep the remainder in a bag from the supermarket, squeezing out the air first.
We have one of those too. Looks like a big purple onion.
I read somewhere that you can decrease the onion smell by dicing it, then soaking the onion in water for 10-15 minutes, then draining the water and drying the onion as best you can. I’ve tried that and it seems to help, both with white and red onions. I would cut up red onions before and the smell would be pretty strong later when I’d open up the fridge. Now it’s not so noticeable.
I’m a sandwich baggie sort of guy but generally, I don’t have any left over onion to keep.
I did, but not the way my SIL did for us once. Many years ago, she was helping us deal with a huge amount of fresh produce that had been given to us, including several pounds of red onions.
She placed them into the freezer. Whole. By the time I found them, they were useful only as potential murder weapons.
Oh. The question. Zip-Loc bag in the door of the fridge.
I feel confident and encouraged that I can now handle the Onion Issue. Thanks ever so! 
I gotone of those for Christmas 3 or 4 years back, and we use it as well. 
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This. But on the rare occasion that I have half an onion left (like one of those giant purple ones), I just chuck it in the butter compartment of the fridge door, so it doesn’t get lost behind the leftover broccoli that nobody’s really going to snack on. I use butter and onions almost every day, so I’ve never noticed a taste transfer there.
I use plastic wrap or a plastic bag, and if I don’t use it within a couple of days I throw it out. Also, I shave off the cut surface before using it, because a cut onion starts to develop an unappetizing smell after a while.
I seldom need less than a whole onion (or tomato) but, if I do, I wrap the remaining portion in cling wrap. It’s always as good as new when I need to use the remainder.
I never use less than a whole onion. Never. I buy smallish onions to make this more easily possible.
However, my wife will religiously only add half an onion to any recipe that calls for a whole one, so if I know she’ll be cooking, I buy really big onions (she’ll add half an onion, regardless of the size of the thing). She chops the remainder and freezes it in a small airtight plastic box.
If one’s available, we put the other half, usually uncut, in a tupperware type container and throw it in the fridge. If the container isn’t available, a baggy works too.