Six on the wooden flute and I use them all, surprisingly often. I would dearly love C and C# keys below bottom D.
Since my progression was Highland pipes > uilleann pipes > keyless 6-hole flute > keyed flute, I never had any issues fingering a flute with all its holes in a straight line, because that’s how pipes are.
What messes me up are so-called “ergonomic” flutes and whistles that have their holes all over the place. Can’t do it.
About the number of keys, were I to get into fluteplaying now, and ordering a new bespoke flute, I would probably get just Eb, one F (either one), G#, and Bb. With those four keys I can play anything, pretty much.
Some might say “what on earth would you use the Eb key for?”
Being a piper, I would use it to throw in “the ghost D” every now and then, which I know isn’t traditional fluteplaying. But mainly it’s for playing non-trad music.
Muineira Sul Sacrato Della Chiesa, Battlefield Band setting (starting on E), of course! 
I have no keys on my flute, and I use all of them.
My main flute is keyless, so I half-hole or cross-finger as necessary. I can do this more or less satisfactorily for everything but Eb.
I do have an 8-key, but it’s a smaller-holed flute with not much “bang”, so usually I use it for gentler stuff - O’Carolan tunes and the like.