Don't forget tortoiseshell butterflies use nettles on which to lay their eggs. I watched one yesterday laying on every nettle in a patch. Depending on the day, you can tell they're there because the leaves are a bit curled round, and become more so, held together with a cocoon. leave patches for the butterflies and you will be rewarded with having plenty in your garden all summer

Otherwise, yes digging out with the whole root system, which is yellow, is the only way. Mowing is second best if the digging is too hard going.
Never let them flower (light green tassels) and seed or you'll never get rid of them. We have had to abandon a couple of compost heaps where nettle seeds got in. I too have tingly fingers today from pulling small nettle seedlings out of plants potted up in home made compost.
Nettle tea for plants and people, spring vegetable for a spring pick-me-up, gently steamed, mulch around plants, add to the compost heap, as long as there's not a single seed set, are all possible uses, as WBF says. I mix nettles with comfrey to make liquid fertiliser which is well balanced, as nettles provide nitrogen and comfrey supplies potassium - it's a horribly pungent, stinking concoction while it's infusing, so leave it well away from you.
Sheep, at least primitives, like to eat the older plants at the end of the season, but don't seem too fond of the younger ones.