Whatever lawn grass already grows well in your yard is usually the right answer. UT Extension lists tall fescue, fine fescues, and Kentucky bluegrass as Tennessee's main cool-season lawn grasses, and bermudagrass and zoysiagrass as the main warm-season ones. Tall fescue handles part shade better. Bermudagrass and zoysia want full sun.
All of them have fine, fibrous roots, not the woody, pipe-seeking roots of trees and shrubs, which is exactly what you want. Grass holds the thin soil cover in place, lets the field breathe and dry out, and makes a wet spot or bright green stripe easy to spot. If the field is bare, reseed by hand with the same mix as the rest of the lawn rather than bringing in a tiller or a load of topsoil.
One mistake to avoid: watering and fertilizing the field to make it match the rest of the yard. Grass color alone cannot tell you whether a field works. Drought can brown the cover; unusually lush growth with wet soil, odor, or slow drains needs investigation.