The old books recommend Duco cement, but nobody I know uses this anymore and I don't even know if it is still on the market. You will probably use at least two different sorts of glue. One to hold the specimen on a pedestal, and the other to mount the pedestal in the box.
Be aware that various trade names complicate and obscure what should be a simple issue. For example "Gorilla Glue" was once a name for a brown polyurethane glue (that you would be unlikely to use for micromounting). Now "Gorilla Glue" is a company that sells a variety of products with different characteristics, including a Cyanoacrylate "super glue". The same is true for Elmers, Loctite, Weldbond, and other products -- so pay attention.
I now use a permanent marker (sharpie) to date all my bottles of every kind of glue. There is no sense in being cheap and using glue that has significantly degraded.
I particularly like "Aleene's original Tacky Glue". This comes in a gold colored bottle and is sold in craft stores. It is thicker than regular white glue, which makes it very easy to work with. It can be removed by soaking in water,
I also use and very much like "Weldbond". This is thicker than most PVA glues, but not as thick as "tacky glue". People mix it with water and use it for a sealer on porous surfaces. I use this when I am joining two flat surfaces that mate nicely.
Tacky glue is what I used most of the time to mount minerals on corks or pedestals. I use Weldbond when the glue joint will be thing, such as when mounting a speciment with a sawn surface.
I have tried using Weldbond to attach paper, wood or cork to plastic. This does not work well. The glue forms a flat surface that pops off the plastic readily. Use CA glue instead (see below).
A particularly nice thing about PVA glues is that they are water based. This means no fumes, and it also means that you can remove things like labels by soaking a micro box in water. The FAQ on the Weldbond website says: "Weldbond can be quickly dissolved with acetone. Because Weldbond is water based it can also be dissolved with a soaking of, or submersion in water. This will dissolve a bond."
I intend to make tests and find out how hard it is to soak off a well cured label attached with Weldbond. They say Weldbond has a 5 year shelf life and that if it thickens up, you can just thin it with water.
One person I know got a gallon of the original Elmers, along with some lamp black and had it mixed at a paint store so he now has black glue, which is not a bad idea.
This is perhaps the best way to attach a cork or wood pedestal to a polystyrene micro box. The glue partly dissolves the plastic making an extremely strong bond. Unless you use an absurd amount of glue, this would never be noticed though.
I am now using "Loctite Super Glue professional liquid" that I get in a 20 gram bottle at my local hardware store.
Another fellow I know recommends "Rhino Glue", which is only avaialable online (either from Amazon or Rhino). As near as I can tell this is just a brand name for Cyanoacrylate.
I did some tests gluing 3 corks onto the black bases for the Euro-style boxes that I now use. All three broke through the cork (when I pushed with my thumb). The break was just above the glue bond, which remained intact. This was true even with year old CA glue that had thickened significantly. Cleaning the plastic with alcohol seemed to yield no benefit, and is just wasted time.
I have a bunch of mounts on paper liners in the older micro boxes. I plan to cut the "ears" off the liners, then glue these paper squares onto plastic bases from the "euro" style boxes. My tests indicate that CA glue works very well. 4 dots on the paper bonds it securely to the plastic. White glue does not work for this purpose.
Not many people use this, and it is all but impossible to remove once it cures. You could of course view this as a virtue, but most people don't. One fellow I know uses this exclusively to mount his mineral pieces on pedestals. Another fellow I know curses him for doing so.
The clear Gorilla glue variant does not expand, but also is said to not work as well with the water soluble minerals.
Tom's Mineralogy Info / tom@mmto.org