Horse, alpaca and sheep manure tends to be bulky but not too strong so it's good for general fertility and adding humus to your soil.
I don't use pig muck at all because it tends to have heavy metals in, but my brother swears by it

. I would imagine it would fit in the low bulk, high nitrogen bracket.
Chicken manure is very concentrated so needs to be really well rotted so there is no sign of the bedding left. It doesn't add much bulk but does add plenty of nitrogen. If the poultry has been bedded on wood chips then the high nitrogen content will help to rot it without robbing the soil. Ours are bedded on straw.
I don't have the luxury of choice beyond sheep and chicken and I tend to rot the two together for a general purpose FYM.
Garden compost is very different and I do try to have separate heaps for that. Veg such as carrots etc don't like strong manure so I use manure-free compost on those areas.
Some veg are much heavier feeders than others, so I would add more FYM to those crops - brassicas, potatoes and the like. Leguminous crops don't need much nitrogen as they make their own, but they do appreciate plenty of bulk, so Horse manure, as long as they have been bedded on straw not woodchips, would be ideal.
Tomatoes, squash and other plants which produce a 'flower' crop, need high potassium rather than high nitrogen, so benefit from the addition of something like comfrey or borage.
That's all I know. I tend to try something and if it works I stick to that, if it doesn't I try something different.