Many vines, devoid of support, will spread across the ground to act as a cover. Consider Virginia creeper (Parthenocissus quinquefolia), chocolate vine (Akebia quinata), trumpet vine (Campsis radicans) and climbing hydrangea (Hydrangea anomala subsp. petiolaris). One of my favorites, winter jasmine (Jasminum nudiflorum), blooms in my garden in January and early February, and even saves some blooms for April. It roots where the nodes on the underside of the leaves come into contact with the ground, thereby producing inexpensive offspring. Take note, however, that some of these vines can become pesky (for example trumpet vine) and can pop up from root runners where you least expect them.
Concern about groundcovers running amok and overtaking the garden is certainly warranted. Goutweed (Aegopodium podagraria), lily-of-the-valley (Convallaria majalis), bracken fern (Pteridium aquilinum), chameleon plant (Houttuynia cordata), stonecrop sedum (Sedum acre), silverweed (Potentilla anserina), ribbon grass (Phalaris arundinacea) and blue wild rye (Elymus glaucus) can spread readily, killing other plants in their path. There may be places where you can take advantage of these rapid spreaders, however, such as an area with exposed road cuts, a site with severely compacted soils, or a space under the skirts of spruce trees. Be warned though – where these plants are not wanted their removal can be challenging, since root pieces and seeds left behind readily produce offspring. At cottage properties in particular, groundcovers such as Japanese spurge (Pachysandra terminalis), vinca and various ivies can quickly make inroads into forested areas to the detriment of the native flora.
Prairie smoke make a terrific groundcover bordering a path or edging a border.