Creeping Jenny- Golden Moneywort. It stays very low- 2" max. It is very invasive, but does not climb into other plants.

Golden moneywart turns very yellow gold in sun and just slightly greenish in deep shade. Either way both the colors are appealing as long as the plant colors nearby do not clash.

I have Lamium with light pink flowers around my hostas as well as Sweet Woodruffs. They look great right now, especially the Lamium that appears luminous in the shade.

I 2nd the lamium suggestion - I have it growing with my hostas and they co-exist beautifully. Some find it invasive, but I've never had a problem with it. I'll admit that it's a fast spreader, but it's easy to pull up if it goes where I don't want it to. And there are loads of choices when it comes to foliage color.

I would not use any groundcover at all, but use leaf mulch.

On second thought, while inspecting the garden this morning I have oak fern, Gymnocarpium dryopteris, covering some large patches.

Moss

I use Ajuga under my hostas.

Here is a link that might be useful: Ajuga

Violets - I have both the scented (pink ones and purples ones too) and then the regular scentless purples, pinks, whites, yellows. Yes! they spread, even in my lawn, but they are very pretty especially when in bloom. The rest of the year, you have crowns of fresh looking green leaves.