I love sedums...takes a lot of sun and little maintenance. They really fill out beautifully in the garden.

two groundcovers that'll grow anywhere are snow-in-the-mountain and Chinese lanterns. Both are VERY invasive, but if you got a spot where absolutely nothing will grow, try these, they spread like crazy and both will flower

I can personally vouch for Acre and Dragon's Blood sedums. I have dragon's blood sedum in a small hypertufa container I made a while back and it looks fantastic. There are so many kinds of sedums that can add different textures and year round color (at least here in the PNW) to the garden. I have made little miniature sedum gardens as vignettes, gifts and little "fairy" gardens with the kids. Good drainage and some sun love is all they need--great for the beginning gardener and so delightfully tactile.

I also have many kinds of creeping thyme, but the fuschia/intense red-violet one is one of my faves.
Some varieties of thyme are great at weed suppression around larger perennials, too.

I have to second all the sedum-love here (or third, I guess)-- I don't have particularly good drainage in one spot and yet the low growing sedum do fantastically well. Highly recommend it.... ours has a sweet small white flower and they get tromped on by kids and dogs on occasion and hold up just fine. Lots of sun a necessity though!

I bought a Sedum anglicum 'hartland' for a tiny potted succulent garden and it is delightful. It would be wonderful to have it spread across a swath of ground outdoors.

I also really enjoy Senecio rowleyanus or "string of pearls" which is often seen in hanging baskets but works really well over rocky surfaces as cover.

Groundcovers are very essential to your garden, but you'd have to comb them for snails time to time. It is a perfect hiding spot for them.

You missed my favorite! Creeping phlox - bloom in the spring, great vibrant colors - maybe it's an east coast thing.

rookie question here: I have a concrete backyard with two squares cut out for two plum trees. Would it be ok to plant groundcovers around the trees? Currently the ground is covered in...weeds and a few nasturtium plants. I'm afraid of the groundcover suffocating my trees.

@benayse: Most groundcovers won't harm your trees. Just don't plant anything that climbs. Check out what's native in your area, there are many good choices. Also consider planting moss, since the space will be shaded by the trees.