Abrasives today are, more and more, precision-manufactured substances, but natural mineral abrasives still have a strong position. A good abrasive mineral is not just hard, it's also tough and sharp. It also must be plentiful (or at least widespread) and pure. Not very many minerals share all of these attributes, so the list of abrasive minerals is short—but interesting. Let's start with the sandpaper minerals.
Sanding was originally done with (surprise!) sand, that is, fine-grained quartz. Quartz sand is hard enough for woodworking (Mohs hardness 7) but not very tough or sharp. The virtue of sand sandpaper is its cheapness, but I've never seen any and I doubt that it's made any more. Fine woodworkers do occasionally use flint sandpaper (glasspaper), though. Flint (a form of chert) is a rock made of microcrystalline quartz. It's no harder than quartz but it's tougher, so its sharp edges last longer. Garnet paper is still widely available. The garnet mineral almandine is harder than quartz (Mohs 7.5), but its real virtue is sharpness, giving it cutting power without scratching wood too deeply.
But the workhorse abrasive of sandpaper is corundum. Extremely hard (Mohs 9) and sharp, corundum is also usefully brittle, breaking into sharp fragments that keep on cutting. It's great for wood, metal, paint and plastic. All sanding products today use artificial corundum, aluminum oxide. If you find an old stash of emery cloth or paper, though, that probably has the real mineral: emery is a natural mix of fine-grained corundum and magnetite.
Visit Woodworking Guide Chris Baylor to learn more about choosing sandpaper. He includes the various artificial grits that have never been minerals.
Three natural abrasives are commonly used for polishing and cleaning of metal, enamel finishes, plastic and tile. Pumice is a stone, not a mineral—a volcanic product with a very fine grain. Its hardest mineral is quartz, so it has a gentler action than sanding abrasives. Softer still is feldspar (Mohs 6), which is most famously used in the Bon Ami brand of household cleaner. For the most delicate polishing and cleaning work, such as jewelry and fine crafts, the gold standard is tripoli, also called rottenstone. Tripoli is microscopic, microcrystalline quartz that is mined from beds of decomposed limestone.
Sandblasting and Waterjet Cutting
Applications of these industrial process range from scrubbing rust off of steel girders to inscribing gravestones, and there's a wide range of blasting abrasives in use today. Sand is one, of course, but airborne dust from crystalline silica is a health hazard. Safer alternatives include garnet, olivine (Mohs 6.5) and staurolite (Mohs 7.5). Which to choose depends on many factors besides mineralogical ones, including cost, availability, the material being worked, and the experience of the worker. Many artificial abrasives are in use in these applications, too, as well as exotic things like ground walnut shells and solid carbon dioxide.
The hardest mineral of them all is diamond (Mohs 10), and diamond abrasive is a large part of the world diamond market. Diamond paste is available in many grades for sharpening hand tools, and you can even buy nailfiles impregnated with diamond grit for the ultimate grooming aid. Diamond is best suited for cutting and grinding tools, though, and the drilling industry uses lots of diamond for drill bits. The material used is worthless as jewelry, being black or included (that is, full of inclusions) or too fine grained. This grade of diamond is called bort.
The powdery substance composed of the microscopic shells of diatoms is known as diatomaceous earth or DE. Diatoms are a kind of algae that form exquisite skeletons of amorphous silica (learn more at the UCMP site). DE is not abrasive to humans or metals or anything else in our everyday world. But at the microscopic scale, DE is very damaging to insects. The broken edges of crushed diatom shells scratch holes in the hard outer skins of insects, causing their internal fluids to dry out. It's safe enough to strew in the garden or to mix with food, such as stored grain, to prevent infestations. And when they aren't calling it diatomite, geologists have another name for DE, borrowed from German: kieselguhr.