I would suggest alpine strawberries. For 2 years now I have used them as a low border for my mixed bed of roses, annual/perenials. They reseed but are easy to remove or transplant, the slugs and birds don't bother them. Even my army of voles leaves them alone. I eat some of the berries-they are great-but if not they are very attractive. The foliage and flowers are very attractive-unlike regular strawberries. You can buy plants or start from seed so quite inexpensive.

Thank you, quenz.
Do you know the name of your alpine strawberries? I see online several names. What do you do with them in fall?

I have Fragaria 'Lipstick' with pretty pink blooms intermittently most of the year. I'm in a warmer zone so I don't do anything with mine in the fall and they are evergreen here. They have spread from a small driveway circle bed out into the gravel driveway where they don't even seem to mind being run over. And as quenz said, are easy to pull up from where they aren't wanted.

I have several different kinds-some from seed. My favorite is a white berried type-tastes very much like pineapple. And all of them are highly fragrant in fruit. They are pretty much evergreen-I did shear some back in very early spring to tidy up.

How about some saxifrages?
London Pride, solid green or the variegated one. There are a lot if different Saxifrages with various leaf shapes and colors.
But some have runners-- S. stolonifera--so you may not want that type.

http://www.inlandnwgardening.com/gallery2/d/1778-2/0704359.jpg

Here is a link that might be useful: London pride under shrubs

Thanks to all of you. I like both ideas. Will think about them.

Nice suggestion, plantknitter :-) I am very partial to London Pride, especially the variegated form. This is a plant that seems to be able to grow under just about any condition, other than very wet, and I find it to be a great evergreen carpeting plant for even dry shade. The foliage always looks good, it doesn't seem to be bothered with pests and it's stunning in bloom with those airy little pale pink flowers.

I have to remember to add it to my shopping list of plants for my new garden :-)

I like Saxifrage as a ground cover too. A hardy sedum also works really well or my one of my favorites: Epimedium. Most Epimeduims provide year round interest and require no care, other than lifting and dividing every five years or so. They're a bit pricier than the others listed here but well worth it to me. I have found the Fragaria to be a bit of a thug in my garden and have been trying to get rid of it in one bed for the last 6 years, with bits reappearing every year. I don't know if its just the variety I have that's a spreader or if it just likes where its growing.

Epimedium. This can be nice. Madrone, what kind are is your favorite?

Heucheras are evergeen. They have big bold leaves and tiny roots that rarely spread.

Epimediums are happiest in shade, if this is what your planting area offers. And some are very happy in dry shade but most will not be as happy in a sunnier position.

Local nurseries typically offer a selection of the more common ones - E. grandiflorum, rubrum, x versicolor, etc. but there are many more species, hybrids and named cultivars available through specialty nurseries like Naylor Creek. I just got my 2010 catalog for Naylor Creek and immediately earmarked several new epis to add to my shade garden :-) And they definitely tend to be more expensive than your average perennial - sometimes a LOT more expensive.

And I'd not exactly consider them a "tiny" border plant. Depending on variety, the foliage of a healthy clump can reach 12-15" tall and they can form quite wide plantings over time. But they are pretty much maintenance free, only requiring an annual grooming to remove old, tattered foliage (much like hellebores).

Here is a link that might be useful: epimediums