At the risk of sounding too snooty, I find that wooden boxes with bored out (glued in) inserts into which the shank of the cutter slips nicely work the best. If I drill the hole for the shank of the cutter 1/32'nd of an inch (call it 0.75 mm) oversize, then the cutter stays a nice slip fit even after I have applied a good oil (I prefer tung) finish. A bit of piano hinge and a 1/2-turn "Draw-Tite" latch makes it complete. A couple of "grips" glued to the sides made out of 1X2 timber (net 3/4 X 1-1/2 inch) with a quick undercut on the bottom edge finishes it off and makes it easy to carry.
About 30 years ago I built a roll-away with drawers made in that fashion, but I had to move into "less spacious" quarters and I was offered enough $$$ to part with it. My replacement is (back to) boxes that fit nicely on shelves. I organize my cutters by general type (HSS, Carbide, Ground-for-Steel, Ground-for-Aluminum, etc.) so I can find my desired cutter without dragging down and opening a dozen different boxes.
I keep going back-and-forth on tap storage. At various times I have grouped taps in their own "stand" (i.e. a block of wood drilled for the tap set (bottoming, taper, spiral, and reduced shank, with a hole to hold the tap drill) -- and then go back to boxes that have grooved storage for a class of taps (bottoming, taper, etc.) that holds a range of taps. I have not really decided (even after 4+ decades) on which approach I like. However, as I don't throw out the pieces, swapping back and forth is (almost) painless.
