Answer: The most important thing is to match the sun exposure to the groundcover. It wasn't clear from your description, but here are some possibilities for shade or sun. Good luck!
Here are a few groundcovers that take shade and should tolerate your cold winters: Vinca (periwinkle), partridgeberry (Mitchella repens), English ivy (Hedera helix), and bunchberry (Cornus canadensis).
These take sun: Wooly yarrow (Achillea tomentosa) makes a flat mat of deep green hairy leaves with golden flowers on 6-10 inch stems in summer. Takes full sun and moderate water. Carpet bugle (Ajuga reptans) spreads quickly with runners. It has flowers on 4 inch spikes. Many varieties are available. However, it is susceptible to rot if drainage is poor. It takes full sun and regular water. Arenaria is often used as a lawn substitute. It has small white flowers in late spring and summer. It takes full sun and regular water. Phlox subulata (Moss Pink) reaches about 6 inches high. It blooms in late spring or early summer. There are many varieties available. Evergreen candytuft (Iberis sempervirens) is 8-12 inches high with shiny dark green leaves. There are many varieties with different bloom times. Some flowers are tall enough to use as cut flowers.
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