The only hays I've heard of being fed are: alfalfa, timothy, meadow, orchardgrass, oat, and recently bromegrass. The most popular hays are timothy, orchard and alfalfa I'd say. I give a mix of timothy, orchard, meadow and sometimes alfalfa as a treat. I just bought all those because I wanted to find out what hays my chin likes so it's not like you have to feed them multiple kinds.
I have no idea what the nutritional requirements are and why certain hays are fed more often. The hays I feed are available to me and widely available to others. I don't know which hays are unsafe to feed.
I have attached a pdf of one of my labs in class on forages. It has timothy, alfalfa, orchard, brome, meadow and oat hay partial nutritional content (on average). The nutritional analysis is on the first two pages. I honestly don't know where these data came from but I'd assume they are correct, since my teacher gave them to us.
DM= dry matter (nutrients are listed on 100% DMB and hay is about 90% DM), CP= crude protein, NDF and ADF are both components of fiber, but the lab has the crude fiber listed, and calcium, phosphorus and vitamin A of each hay. The CTTL TDN is just the total digestible energy of the hays for cattle, so ignore that column as well.