An oversized shirt is the thing you put on when you want to look like you've made an effort without doing any of the usual work. It covers a multitude of sins around the waist and hips, it layers over most things, and it does the job of a jacket on days that aren't quite cold enough to justify one. That's the whole point of it. Anything beyond that is a bonus.
The difference between a good one and a bad one is mostly in the shoulder and the fabric weight. A well made oversized shirt has a dropped shoulder seam that actually sits on your upper arm in a deliberate way, not a collapsed one, and the fabric has enough body to hold a shape rather than go limp by lunchtime. Cheap cotton blends crease into a map of where you've been sitting and stay that way. Check the buttons too, plastic ones that are too thin will crack at the edges within a season, and a placket that isn't interfaced properly will gape between buttons when you move your arms, which on an oversized shirt happens constantly because there's so much fabric to move. Run your hand down the seam at the side. If it puckers, that's not going to iron out, that's how it's made.
If you've got none, start with a plain cotton one in white or a pale blue, nothing printed, nothing with a logo. It's the version that does the most jobs. Mine sits a good four inches past my wrist in length of sleeve and I roll it, which is partly style and partly necessity, but on a regular fit it would have stopped at the bone and looked like I'd borrowed it from someone the right size, which is a different look entirely and not the one I'm after. For pairing, it wants something fitted underneath to balance all that volume, a vest, a slim trouser, a pencil skirt even, something that gives the eye a line to follow so the whole outfit doesn't read as one shapeless block. Tucked loosely at the front with the rest left alone is the easiest way to wear it and still look like you decided something, rather than just not finished getting dressed.
In the shop, hold it up against yourself before you try it on. The shoulder seam should land somewhere past your actual shoulder, usually an inch or two down the arm, and if it's falling further than that it's not oversized, it's just too big, which is a different problem entirely. Check the hem too, a lot of these are deliberately longer at the back, which is fine, but on me that back hem can end up round my calves rather than my hip, and I've learned to check it sitting down as well as standing, because that's when a long back hem really makes itself known.
Reach for it on the days you want to look pulled together in under two minutes, over a vest with jeans, or open over a dress when the weather can't make its mind up.