When we moved in there was a small patch of grass on the south side of our house next to the porch. It was cut off from the rest of the lawn by the driveway on one side and the back patio on the other. My husband would mow it, but the grass would grow back much faster than the rest of the yard. He figured it had been planted as a garden of ornamental grasses...I wasn't so sure. After doing some research online, I discovered that we had a patch of Yellow Nutsedge - not a plant I wanted to keep around.
In the fall, I took a shovel and dug up every inch of that weedy grassy patch. Afterwards I carefully dug through the loose dirt pulling out all the nutlets and rhizomes I could find, then I covered it with a thick layer of newspaper and waited for winter. The next spring I still had some sedge to pull up, but eventually it stopped coming back and I was able to plant my herb garden in it's place.
In Zone 4, most of the herbs I plant are considered annuals, but I'll take perennials when I can get them! Chives were the first to go in and the least fussy. I'm convinced they'd shoot up through the snow if the sun could get at them! I planted three different kinds of mint in pots (to prevent them from taking over the garden) and kept them in the garage for the winter. They're all starting to come back, although next winter I'll have to remember to give them a little water from time to time. I also planted lavender, sage, oregano, and a border of creeping thyme, which I'm happy to say have all survived their first Minnesota winter! I covered them with a thick layer of leaves in the fall and we had good snow cover the entire winter, so that probably helped.
Since so many of my annuals survived, I'll just be starting basil, chamomile, and cilantro from seed. I can barely wait to make my first batch of pesto!