Since I only cook for one or two, I will tend to buy an onion and only use part of it, and then keep the rest in the fridge. Sometimes it will be a while before I use the rest. Is there any guidance on when I should finally just throw it out? What are the signs to look for? What degradations should I expect to trade off against constantly throwing out half an onion (decreased flavor, increased bitterness)?
When I use fresh onions, we store the unused parts for up to a week in our fridge in either a ziploc or a sealable rubbermaid-style container. For particularly pungent onions they go in the crisper drawer to keep the smell from being too strong in the rest of the fridge, but usually they're just on one of the shelves. Stilltasty says 2-3 days but my personal experience says otherwise.
Generally in our house, they don't last long enough for the flavor to degrade too much, but they will get a little dried and the smell will get stronger over time. After a week, they shouldn't be bad in the sense that you'll get sick - but you'll want to use them in a meal where the onion is the star, as the aroma will take over the dish.
Another option that I'm a big fan of is to dice or strip the leftovers (or, a 5 lb bag at once!) and freeze them on a cookie sheet (instead of in a bag, where they'll turn into a giant clump). Once fully frozen, dump them into a plastic freezer bag, and store in the freezer for easy access to a handful at a time. For most applications, you can defrost them right in the pan/pot/casserole as you cook, so they're very handy and accessible. We do the same with peppers in our house for convenience.
I've had good luck with sealed glass containers ... including when storing sliced or diced onions. I find that plastic containers can start to pick up the onion smell if you leave onion in there for too long.
I've kept a halved, but otherwise in one piece, onion for over a week. If it dries out a little, you can always just take a slice off the cut side.
Cut up onion can keep for about a week, but it helps to have used a sharp knife -- a dull knive damages more cells and it'll start to break down faster. (and if it's sharp enough, you can slice it up, but keep the shape, so there's no extra exposed cut surface to lose moisture from). The more damage you do to the cells, the stronger the 'oniony' smells will get.
My judge on if it's still okay to use is to look at it -- if it's gone translucent, smells funny or looks funny (including dried out), I pitch it.
Another option is to caramelize one or two onions in a frying pan, and then freeze the pre-cooked onions in smaller portions. (I tend not to like cooking with frozen onions, since the extra water will steam the onions rather than allow them to cook at the higher temperature you could achieve without the extra water.
A couple of years ago I found a container designed to store cut onions. It looks like a plastic bowl with a rubber and metal lid. The smell is kept inside the container and the onion lasts several weeks.
I don't know how it works, but it most certainly does! Before I had it, I would wrap an onion in plastic and it wouldn't last quite a week.
Cut onion can be stored 7-10 days in a sealed container in the refrigerator. I've not experimented specifically with glass or plastic, but I can understand how glass would be better. I'll have to give that a try. Sharp knives are key, that was a spot on tip!
With regard to the comment about cut onion absorbing bacteria, that is false. Anyone interested in more on that can click here: http://onions-usa.org/faqs/onion-flu-cut-myths
straight from those who know what we are crying about..
I also used to save unused onions for another day, until I read that a cut onion should not be reused in cooking as the peeled surface drinks up all the toxins and bacteria out of the atmosphere and into the onion.
One remedy for flu symptoms is to place a cut onion on the bedside table before sleeping. In the morning the onion will be black in colour having drawn up all the bacteria etc in the room. And the flu symptoms will have disappeared.
I would rather chuck out a half-used onion (after leaving on the kitchen surface overnight) knowing it had worked as a kind of disinfectant.