Flies that I might readily use go into my fly boxes that I carry with me on the water.
Flies that need to be stored get filed away.

I use the C&F style boxes with the flip page. Each box will hold a bunch of flies.
There are a lot of boxes that are very similar in design. Some more affordable than others.
Each fly box is labeled for what's in it. i.e. "DRY FLIES", "NYMPHS", "SCUDS", etc.

For flies that I file away, I use the large, flat Plano style boxes with bunches of dividers.
I've even made more dividers when I've ran out. I use the 370 or 3700 size boxes. (Same thing, just depends on who sells them.)
I have a tackle bag from one of the big box tackle retailers that will hold a ton of stuff.
It will hold five of those 370 style boxes neatly in a lower compartment. I label the front of each box to keep things easy.
The upper compartment will hold quite a bit of stuff too.
I put leaders, tippet, indicators, mucilin, flashlight, spare fly boxes, etc in there.
It has two side pockets. Each one will hold a slightly smaller 360 or 3600 size box. But I keep leader building material in one pocket.
It also had a flap on the side with little bungee straps sewn in to keep tools. I keep extra nippers, forceps, and nail knot tool there.

The greatest thing is that it's easy to load that bag up and have everything I need in one big, organized bag.

I got it as a xmas gift and I have to say it's been one of the most useful gifts ever.

Brian