I use the white clover in parts of my yard that don't grow grass well, and to compete with the weeds. I love it.
Most grass needs full sun, so you might want to say whether your yard gets full sun or shade or what.

We have a lot of sorrels in our yard, right now they seem to be choked back from our grass (and I suspect due to the crazy dry spell we are in) but normally they pop up in our yard in large clusters and have pretty little pink flowers on them. This year we even had some yellow ones come in. (you will need to mow your sorrels just like with grass)

Beware though, sorrels can be VERY invasive. If you decide you want them gone, they can be a nightmare. They are a type of bulb, and because their stems are so tiny, 9 times out of 10 they bulb will break off if you try pulling them. We have some in beds that we needed to get rid of and the only thing I have found so far that works is Spectracide Weed and Grass Killer.

There is also Magic Carpet Thyme, you can find it at Michigan Bulb.com under ground covers. It normally is used for walkways so it holds up well under heavy traffic.

I don't have any grass and probably have over 300 plant species on my postage stamp streetcorner. For easy groundcovers a few suggestions are: asian jasmine, liriope, nasturtium, potato vine or Aptenia cordifolia "heartleafed ice plant" which trails and seems to be happy all the time with very little to no maintenance.

Here is a link that might be useful: Aptenia Cordifolia or heartleaf ice plant

I agree with the clover--I use New Zealand White because it's more drought tolerant than Dutch white. It's tough enough to walk on, cheap to put in, and no mowing--and it's pretty.