The most commonly mentioned plant is sedge. There are many different types of sedge. Some grow in deep shade while others do well in sunnier locations. Sedges look like grasses with one notable difference. "Sedges Have Edges," which means that sedges have triangular shaped stems instead of round or flat stems found on grass plants.
Carex – The Living Mulch! This was how one speaker described the Pennsylvania sedge (Carex pensylvanica). It was described as looking great, suppresses weeds, and adds organic matter to the soil. Other features included a fibrous root system, shade loving, drought tolerant, grows in poor soil, protect tree roots from mowers, and low maintenance once established.
The Missouri Botanical Garden lists a few native sedges to try in gardens. Cedar sedge (Carex eburnean) is described as producing three inch tall "pocket-sized emerald green tufts of hair-like leaves" that form a dense mat in dry, shady places. Oak sedge (Carex albicans) is a "shade to part sun-loving sedge that grows 8-10 inches on dry to moist soils." Do a search for "carex" at http://www.missouribotanicalgarden.org for more examples.
Another lawn alternative is liriope. Also called creeping lily turf (Liriope spicata) this plant is considered a good vigorous spreading groundcover that can be used as a low maintenance substitute for turfgrass in small spaces. It grows 8–12 inches tall and prefers full sun to very light shade. Hardy to zone 4, it is also drought tolerant. As an added bonus, the plant has small light blue flowers in late summer followed by interesting black, bead-like fruit.
Finally, clover is making a comeback as a good lawn alternative. My dad always used to add white clover to the grass seed mix. Clovers are in the pea family, which includes plants with leaflets of three and an ability to pull nitrogen from the air and store it in special root structures.
Those in pursuit of the perfect, carpet-like lawn consider white clover a very difficult weed. This plant has so many positive features, though, that we might consider lowering that stereotypical view of lawns to see something different, but just as beautiful. White clover is an excellent pollinator plant. It makes its own nitrogen so needs less fertilizer. It stays greener during drought, reducing lawn watering needs. Plus, white clover has no serious pests. To assure that clover does not over dominate the grass mow it high at three inches or more.
The University of Illinois Extension website Lawn Talk includes other alternatives to lawn for shady location at http://urbanext.illinois.edu/lawntalk.