WSU:
It really depends on where and when you're fishing for them. The critters are so ubiquitous and versatile that you really need to regard them as different depending on waters you find them in. For example, I know one spot that is a blue ribbon trout stream, and there will be carp staging in the same water from which I catch brown and rainbow trout. Hard to believe, but true. Other waters are practically opaque and if they're not taking from the surface, you're mostly wasting your time with a fly rod. They are opportunistic feeders, so they may be fixated on a "hatch" of cottonwood seeds, berries, or bugs at the surface.
The book Carp on a Fly is a good read, but is mostly concerned with catching them on clear flats in the great lakes.
A mulberry fly can be made by tying a ball of appropriately colored deer hair. A corn fly can be tied similarly. The difference is that you want a corn fly to be tied of more compacted deer hair. A cottonwood fly is basically a bodyless hackle fly of white or light gray hackle.
It also helps to have a spinning rod and some night crawlers handy because there are days when you simply can't catch them on a fly, even though they're breaking all around. They're the most frustrating fish I fish for.
Carp are underrated as a game fish. I've seen how quickly they can destroy a lake, so I don't like them to be introduced to new waters. After they're there, however, we may as well get used to the idea that they're not going away. Fishing for them is fine sport.
