My favorite thing about crocheting a bag is that you can put zero thought or effort into it and still get a very nice bag. I appreciate that a lot, it takes the pressure off. Things like socks or hats kind of need to be specific sizes and shapes to be of any use to anyone, but a bag will always work as a bag.
That said, I have crocheted so so many bags this month (and lots in general) and I have learnt some interesting things ! Of the bags that I crochet, they tend to fall into two categories: bags that are supposed to stretch, and bags that aren't. Please note that this is certainly not the bag classification system anyone else uses, and everything below is my own opinions based on my own experience and research. I certainly don't know everything (or even all that much, in the grand scheme of Bag Knowledge) about crocheting bags.
This can be anything from pouches (dice or otherwise), to project bags, purses, backpacks, etc. These can be stretchy, but mostly their purpose is to keep whatever is inside, inside. This means mesh is usually out, as mesh can allow things to slip through very easily. I tend to choose more solid stitches like single crochet, half double, or double crochet (US terms)/double crochet, half trebel crochet, or trebel crochet (UK terms). I also choose hook sizes that will yield a thick fabric with few visible holes in the stitch pattern. You can use almost any material for non-stretchy crochet bags. Cotton or linen can produce a beautiful bag, as can wool--although wool has the added advantage of being feltable, which means even denser fabric. If you want a thick bag that can stand up to some abuse (while shedding water at the same time), felted wool is a great choice. You can certainly also make bags from acrylic or other non-natural fibers, but they won't be as durable and won't wear as well (and of course acrylic releases microplastics into the environment, while things like rayon and bamboo* are made using highly toxic chemicals, even if they do biodegrade).
Some considerations for making non-stretch bags:
1. Does it need to keep things organized, or does it just need to hold stuff ?
---------If it needs to keep things organized, consider interior or exterior pockets, or designs that include various sections inside the bag. Otherwise, one large interior is usually fine.
2. Do you need to be able to access the contents very quickly, or it it fine if it takes several seconds ?
---------The easiest bag to access will be the bag that has no closure at all. I have a small felted bag that has no way to close it, and I can just put my hand inside. On the other end of the spectrum is a bag I made to hold projects, and it has a pouch attached for crochet hooks and dpns. Those both fall out very easily, so the pouch must be folded and buttoned to close, and unbuttoned and unfolded to open. In the middle are things like drawstring bags, which require only the drawstring to be moved.
3. Are you going to put things inside that might want to poke through the fabric and fall out, like pens, sewing needles, double pointed needles, or crochet hooks ?
---------Consider that the entire bag might not need to be poke-proof, just the area where you intend to keep pokey items. A small pocket inside that has been reinforced in some way to prevent things from poking through is always a good option. Exterior pockets can be great as well--even better, you can make detachable pockets to move from bag to bag. If the entire bag does need to be poke proof, you can line the bag with fabric, felt it, or use a very tight gauge. If it doesn't need to be poke proof, you can ignore all mentions of tight gauge and dense fabric--just use what sounds good.
4. Is stretchiness a bonus or something you want to avoid ?
---------If you'd like some stretch to your bag (even though its main function does not involve stretching) you can always use mesh stitches or work with an elastic wool yarn. If you're trying to avoid stretch, you could use cotton or linen yarn, or use wool and then felt it. Single crochet(US)/double crochet(UK) can help to make a very non-stretchy bag as well.
By which I mean grocery bags and storage bags, and other such things. I prefer to make stretchy bags with at least some mesh, although not necessarily entirely or even mostly mesh--that really depends on what exactly I intend to use the bag for, and how much yarn I have, what mood I'm in, etc. Stretchy bags can be vastly improved by using wool yarn instead of cotton or linen (honestly I prefer wool for almost every kind of bag) as the fiber itself has stretch, which will help a lot.
Mesh is one of two key ingredients to a stretchy bag. Its powers include the ability to store things outside the bag (if you stuff a fleece into a mesh bag, it will puff out between the yarn but not fall out, thus increasing storage space), the ability to make what yarn you have stretch much further (mesh uses comparatively little yarn to cover the same surface area than other crochet stitches), and just an inherent ability to stretch more than most crochet stitches. The more mesh your bag is, the more you will be able to store in it.
The other key ingredient is yarn elasticity. A cotton crochet mesh bag will stretch some, but not nearly as much as a wool bag of the same design will. This is true of these fibers in general as well; wool is the most elastic natural fiber, and is a great choice for a stretchy bag. This is also a good opportunity to use handspun yarn if you have any that doesn't quite hit the mark for wearable yarn.
Scratchy yarn makes a fine bag; although you may want to use a different yarn for the handle or drawstring.
Underplied yarns can still hold lighter objects (and you may be surprised by their strength anyway).
Overplied yarns can make a good durable bag for especially heavy objects. You can also, of course, remove some ply twist.
Ugly yarns can still be put to use. And a good thing about mesh is that you see more of what's inside than the actual bag, so it's easier to see past the bag itself than you might think.
Super uneven, lumpy bumpy yarns will make a misshapen bag, which is very charming. It can also add a lot of beauty and visual interest.
*Bamboo yarn/fabric can in fact be made through mechanical (rather than chemical) means, but usually isn't. Most bamboo clothes and yarn are made chemically; the mechanical stuff must be sought out specifically. There are two types of chemically made bamboo fiber--bamboo rayon and lyocell bamboo. The rayon is made using the same chemicals all rayon fabrics use, which are toxic and must be discarded after use. Lyocell also uses toxic chemicals, but these can be reused to keep making bamboo fabric, which makes lyocell bamboo more environmentally friendly than rayon, but still far less environmentally friendly than mechanically made bamboo, or any other natural fiber source.*