I now use the "european" style boxes that measure 1.1 by 1.1 by 0.85 inches. I buy these with a thin black base and what I call a clear lid. Many collectors I know use these now as they are both available and reasonably priced. They are sold by Shannon and Son minerals.
I glue my label to the black base and store my micrmounts "upside down". The most important aspect of this is gluing the label onto the same part of the box as the specimen, making them inseparable. If you put the label on the lid, things are certain to get accidentally swapped around.
I have never had a problem with mounts falling off, but I am sure it can happen. I have actually seen more micros stored right side up that have had glue bonds break and ended up rattling around in the box. The piece may get damaged, though this seems to be rare. Usually you discover the problem and just glue it back again.
Some people like to open their drawers and see all the minerals through the clear lids. Such people often add a small auxiliary label to the lid. I consider this unnecessary extra work. To me the lids are just dust covers and could get moved around at random. This is certainly a matter of taste, but consider that these are micromounts and in most cases won't be interesting to look at without magnification. Back when I used the smaller "standard" boxes, I bought them in all black plastic and never worried about seeing the mount through the box.
I find it ideal to open my drawer and scan across rows of labels rather than fragments of rock, but to each their own.
You used to be able to get the "american style" boxes from Althor Products in New York. They still sell these (55 cents when you buy 1000 or more). Look for the P-1 box in the "friction fit" section. It looks like only the all clear variety is available. They uses to sell black paper liners for these and maybe they still do. Long ago I called them long ago at 1-800-688-2693 and talked to Kelly Anderson. They were nice people.
All of these leads the the issue where you need to take care that a given box and lid stay together as a pair or all bets are off.
Back in the day when I used the "Althor" style P1 boxes I used them in the all black form. Many people preferred a black bottom and clear lid, but being different materials, the fit was often questionable for these people. A black lid on a black lid was generally a good fit. If you go with an all clear box (which is all that Althor seems to sell these days) you will need to pain the bottoms (a lot of work) or track down (or make) paper liners. Paper liners can warp and do unhappy things if you live where there is humidity. I live in Arizona and don't have those issues.
Uncle Tom's Mineralogy Info / tom@mmto.org