Never thought about cutting up flies, but I have to admit, I catch 80% of my fish on about 10 flies.
Adams (various sizes)
Griffith Gnats (#20, #1
Hairy Weenies
maribou tail streamers (silver tinsel around hook shank and a gob of white or black maribou)
soft hackles (floss or wire ribbed dubbing body)
woolie buggers
poppers (vairous including gurgle pop)
bead head phesant tail (various sizes)
gold ribbed hares ear nymph
"generic dry fly" in various colors (not really a copy of any specific fly, just a generic dry fly that I tie up in a number of different colors, kind of like an adams)
same as above but tied in parachute style
vairous hoppers

That's about it for what I actually catch fish on. Like others, tie a number of different flies just for fun and I fish with them and usually can manage a fish or two with them, but I honestly believe I could cover any water I ever get to fish with the above (maybe a stone fly nymph and an inch worm (green weenie) and some type of attractor fly) and catch fish anywhere I'm likely to go.

This may be the ultimate herasy but I honestly believe size and maybe color are most important. If they're hitting black bugs that are about size #18, and you throw a dark #18 dry fly at them, you're going to catch fish even if it's not a perfect match for the hatch.

Jeff