>> /things/:id/child/:child_id

It seems that nesting isn't super common in my experience. Maybe two levels if completely composite but they tend to be fairly flat.

I see both.

E.g. GitHub /repos/:owner/:repo/pulls/comments/:comment_id

But flat is better than nested, esp if globally unique IDs are used already (and they often are).

Yes but /comments/:comment_uuid that has a parent to /pulls/:pull_uuid is harder to map the hierarchy it belongs to.