For the stumps left behind after felling: a dead ash on the field edge, a row of old orchard trees, leylandii taken out of a boundary. A contractor grinds each stump down below ground level so you can mow over it, re-fence, lay a track or plant again, without digging a crater to get the roots out.
A stump grinder is a wheel of carbide teeth on a small tracked or wheeled machine. It is swung across the stump in passes, chewing the wood into chips, until the stump is well below the surface. The usual depth is around 15–30cm below ground level, deeper by arrangement if something is going to be built or planted on the spot.
The big lateral roots are not dug out; they are left to rot in the ground, which they do over a few years. Grinding the root flare, the buttress roots that spread out around the base, is extra work and worth mentioning if you want a clean surface for a mower or a fence line.
The grindings, a mix of wood chip and soil, either stay in the hole or are taken away. Left in, they settle over a year or so and the hollow needs topping up with soil. Taken away and backfilled with topsoil, the spot can be seeded straight away.
Any time of year. Stump grinding does not depend on the season the way felling does, and a grinder on tracks copes with wet ground better than a tractor. Very soft ground is still worth avoiding if the machine has to cross a paddock to reach the stumps.
Fresh stumps grind more easily than old, dry ones, but there is no rush: a stump cut flush and left is not doing any harm unless it is in the way or starting to sucker.