I had a good chat with Peter Donegan about removing the ash trees. The short version is remove as much of the taproot, as you can without driving yourself mad. The taproot is the main root of the tree, from which lateral roots sprout.

Don’t go chasing the lateral (or surface) roots as the benefit may be minimal. The lateral roots are the ones that fan out to provide support for the tree. These are unlikely to result in new growth so you can leave them in the ground and they should decompose eventually.

If new growth does occur, it won’t matter as the trees I’ve removed were in an area that will be mowed regularly. Any new shoots that appear will be removed before they have a chance to gain much mass.

Don’t take any of this as gospel, it’s what I interpreted/remembered from the chat. All I know is that I’m happy to have them removed.