We use hay (mix & alfalfa) in ours as well as straw & wood shavings. We use more alfalfa & hay than we do straw because that's what we have growing at the farm.
Straw is hard to come by around here (straw is the spent stalks of cereal grains after harvest. So when the combine goes through it threshes out the grain, chucks the stalks back on the field, and after they bale it. The stalks are larger, more mature.)
Hay.. there's a bit of difference in grade/quality which also impacts price. Hay is generally a mix of clover,grasses & legumes. They try to get the alfalfa cut before it goes too far into bud break (has to do with protein content and as well blister beetle management. Btw.. go for first cuttings if you use hay- later cuttings can have issues with blister beetles which are toxic to a lot of animals including chickens.)
Our 8x10 coop is raised and has a wood floor, so we clean it out quite often. We have a layer of pine shavings down and then hay over it. Ammonia is not an issue at all. The birds go nuts playing in the hay looking for bits to eat and pushing it around. (My birds are messy. The only self control the exhibit is they rarely will poop when being held- or in one hen's case- when she hops on our shoulder. Otherwise they will let it rip anywhere, any time, any place.) The benefit to hay over shavings is that the poop tends to fall through into the shavings. It helps the birds keep more clean and essentially insulates the floor.
Carbon levels translate into stink control in the coop. The higher the carbon content, the better the stink control. If you get into composting.. this is where C:N (Carbon to Nitrogen) ratios come into play. Hay has a lower C:N ratio than straw. The fresher the hay.. the lower the carbon levels (it is also why hay bales mold if not cured properly.) We don't have much of an issue with that when it is spread out in the coop as it dries- but in areas where it does get wet/damp.. it breaks down more readily.
Shavings have a really high carbon content. If you garden and get into composting.. it is a C:N ratio of anywhere between 100:1 on up to 500+:1 (The more seasoned/ old the shavings the higher carbon content it has.) Hay is a lot lower (fresh cut hay can be as low as 20:1 on up to 50:1 for an old bale)... straw is roughly 80:1.
Really high carbon content means it takes longer to break down in a compost heap too.
One of our neighbors runs over the fall leaves w/ the lawn mower and dumps the bag into the coop. They have dirt floors though. Come spring it pretty much is a pulverized mess that they strip out the coop and use in the garden.