I have grown Lamium in part shade. Nice leaves, even when it's not in flower.

Vinca minor works for me.
Linda C

I tried this plant this year and it is beautiful and blooms over and over again...I grow mine in partial shade.

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Lamium maculatum 'Beacon Silver'

Geranium maccrorhizum (Big root geranium) makes a very handsome and quite quick to fill in (but never invasive and easy to control) groundcover that's evergreen in my zone. Fuschia flowers now, and I think a white flowered variety is available too. Also, native ginger, (Asarum canadensis), Epimediums, Liriope--all would work.

Sweet Woodruff is really nice. You can see it in this photo on the lower left.

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This garden gets very little sun after the leaves on the trees come out. The plant flowers in spring.

I was just admiring my heucherella (huechera x tiarella) 'Dayglow Pink' in a shady spot under a dogwood. Its blooms are charming and very long lasting.

I also concur with Geranium macrorhizum. I have a variegated variety that is really nice. I have non-variegated one that is a little boring, but functional. macrorhizum/big root translates to low water consumption. I have mine in a spot with tree root competition and it doesn't mind at all.

I'm not familiar with KY zone growing, but I would wonder if some of the earlier suggestions (vinca, lamium, sweet woodruff) might be too aggressive in your climate. I know even here they can be. Some of the lamiums are more aggressive than others. 'Purple Dragon' is almost out of control in my yard, but 'Pink Nancy' in another spot barely gets going. Of course, soil, sun, water plays into the mix of which ones get a little too "happy".

Having said that, if you have a large area to fill, aggressive can be a good solution.

Geranium "Biokovo".

It will grow anywhere, seriously - full sun to shade, moist or dry, it just doesn't care. It also does not get sprawly/leggy like a couple of the suggestions above (lamium, woodruff were lovely the first year for me, after than got sprawly/messy so I yanked them all), yet it does spread/cover at a good clip. Oh, and the scent of leaves is reminisent of Vernors!

Epimedium, ajuga, lily-of-the-valley, wild ginger, plumbago are also good groundcovers in shade.

If it's hardy for you, Mazus reptans may be just what you want. It makes a very low (like one inch) spreading mat of solid green leaves (evergreen too) that spreads steadily, but is not invasive. It has lilac or white blooms that look like confetti in spring and then it will throw blooms off and on throughout summer. The only thing that comes up through it is acorns and a very occasional grass. From a distance it looks like an emerald green lawn. It likes shade, but I have a large patch of it in full sun. It's a very good ground cover.

P.S. I strongly warn you against vinca minor. It is very very aggressive, and is an ideal habitat for SNAKES.