Here are some recommendations for ground covers:
Carex pensylvanica (Pennsylvania sedge): Will take sun to heavy shade and not go summer dormant; can be mowed for a neater appearance. Grows well under oak trees and in dry conditions; 12 inches high by 12 inches wide.
Waldsteinia ternata (Siberian barren strawberry): Beautiful green foliage turns reddish in winter and it has no known problems. Restrained creeping growth allows easy control of margins, which makes it a very effective edging plant. Full sun; 6 inches by 12 inches.
Sedum ternatum 'Larinem Park' (stonecrop): One of the few sedums that likes shade and in fact needs shade. Very attractive and is a good dry groundcover; 2-3 inches by 24 inches.
Thymus doerfleri 'Bressingham Pink' (creeping thyme): Very walkable and covers solidly quickly, unlike many more "stringy" thymes that send out long shoots. This one sends out multiple short shoots from the center, allowing weed-proof coverage more quickly. Sun to light shade; 2-2.5 inches by 18 inches.
Ajuga reptans 'Black Scallop' (bugleweed): Gorgeous, lustrous leaf texture and good solid weed-proof cover. Prefers regular water but takes drought. OK in full shade, but better color and texture with at least three to four hours of sun; 6 inches by 14 inches.
Veronica liwanensis (Turkish speedwell): Beautiful blue carpet in bloom; tight and tough, takes a lot of drought. Likes sun to part shade; prostrate by 24 inches.
Nassella tenuissima (Mexican feather grass): Romantic, wispy texture. Weaves in amongst other plants in a soft and feathery way. Forgiving in terms of conditions. Full sun to part shade; 18-24 inches by 14 inches.
Lysimachia nummularia 'Aurea' (golden creeping Jenny): Can be a bit invasive if in an optimal spot (moist conditions), but in a droughty situation easy to contain and a very good foil for other plants. Full sun to shade; 1-2 inches by 24 inches.
Pratia pedunculata 'County Park' (blue star creeper): Much nicer than normal blue star creeper; deep blue star-shaped flowers and very dense growth. Blooms spring (May-June, but sporadically through summer and even fall). Sparkles! Sun to part shade; 1-2 inches by 12 inches.

Hope this helps!