I'm a magnetic strip user, but only between projects. While a piece is in progress, I lay the tools down in the shavings in front of the lathe with the sharp end toward my searching hand. They go back to the strip when I clean the table.
Storage with nose down means you have to come up with some coding system for the bottom of the tool - too complicated. Storage nose up means you have to isolate them well enough so that you can't accidentally impale yourself if you trip. Strips on the wall near the tailstock are what I finally settled on, because everything else I built filled up with shavings eventually, or was clean, but over away from the lathe.
I do have drawers under the lathe, and they make a great place to store tapers, wrenches, sandpaper, and so forth, with two exceptions. You have to remember to close them or they fill up, and you have to give a quick sweep of the hand on the table over them or curse as the heap of shavings under the toolrest tumbles into the freshly opened drawer. That's why the turning tools aren't in there.