grinds

Flat (or 'V' bevel): good for precision carving in wood and other materials, where a dead-straight/flat cut is important (such as with a woodworker's chisel). A flat bevel, especially a wide one (scandi grind), will help to keep a cut on line.

Hollow: very thin behind the edge, so these grinds excel at fine-slicing tasks (meat, veggies & other food uses, in particular). I suspect this is why you see hollow grinds on a lot of the more 'common' kitchen knives. The thin steel behind the edge makes these easier to sharpen up too (less metal to remove, even when completely re-bevelling).

Convex: great in cutting tougher materials like cardboard, leather. Polishing the convex makes cutting even slicker in these materials, where friction of the blade against material is more significant. Also more durable for chopping tasks.

I tend to separate convex into 'thick' and 'thin' sub-categories also. Even an already-slick thin-bladed knife with a V-bevelled edge can be improved by convexing the shoulders of the bevel, which really reduces friction in deeper/tougher material. A 'thick' convex is better for real heavy-duty use, such as heavy chopping (axes, etc.).

The 'thin' convex is my favorite, overall. Just makes for wicked, super-slick cutting, in most any material. It's durable, and very easy to maintain.
 
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