Do you have a mower? If you do have one then mowing the spearmint and violets may give you the hardy ground cover you are looking for. What doesn't die will grow back shorter and you'll eventually get something presentable.

If you have no mower or dont want to deal with mowing then the best bet will be to spray roundup to get rid of what's there. You could smother it with plastic but that will take a couple weeks (no big deal if you have the time), hand weeding might be more work than you're looking for.

Flagstone would be real nice but costly, cheaper might be pea gravel (softer for kids playing too) a compromise could be a small flagstone area surrounded by pea stone.

I could go on about plant choices but a better idea of what your're planning would be the next step.

Thanks for the reply, kato! Your questions are so helpful in making me think more about this.

We do have a mower but the lawn is so small it takes more time to haul it out of the basement and plug it in than to mow, so we'd like eliminate the need to mow completely. Our neighbors use the mowing method and their lawn looks tidy for, oh, about a week before it needs to be mowed again. My ideal is something that needs tending in the spring and the fall and can be simply enjoyed the rest of the year.

I'm an innocent when it comes to herbicides; I had assumed they were off the table for the lawn-killing project because 5 other yards touch ours and (I assume) you can't control runoff and I don't want to piss off the neighbors. Also, we have lots of snakes living in the yard, neighborhood cats hanging out, and little kids playing back there. All this made me think a non-chemical method was best, even if it was more labor-intensive. But I don't know...

As for replacing the spearmint, I need something that can be swept or raked--the two neighbors kitty-corner on either side have huge oaks that fill our yard with leaves. I am, however, planning a pea gravel "pit" on one side of the yard to replace our sandbox. I was also worried about kids going splat on hard stone (my kids are currently 2.5 and 5.5), but stone just seems like a better choice in the long run. I'm pretty sure I could get salvage broken concrete slabs instead of flagstone to keep costs down, if need be. On the other hand, if we go the groundcover route my initial thoughts were creeping thyme or chamomile. But I'm worried I can't keep my kids out of the yard long enough for the groundcover to establish itself.

I should also say that all the borders of the yard are going to be devoted to fruit cultivation--we already have a serviceberry, blueberry, quince, gooseberry, rhubarb patch, and huge raspberry patch, and are planning on adding blackberries and a fig tree, so the actual area of "lawn-like" area is smaller than 20 x 30.

If time is not a concern, you can smother with layers of newspaper or cardboard. I did this-- took from March till May, though it was a weirdly warm March. Some people feel Round Up is okay/ the mildest of the herbicides.

At our last house, the previous owner had put in landscape fabric and gravel. He liked that it could be cleaned off with a leaf blower. I found it cumbersome to use the blower and after a few years weeds start growing again. You can use Preen and things like that to prevent this, but I personally am unwilling to use Preen. I find that plantings are the best way to prevent weeds.

If it were me and my kids were still small, I would consider deep borders with the wonderful fruits you describe, make a small patio with flagstones interplanted with something suited for the purpose (chamomile?), and have a very small grass area which I cut with a reel mower. I haven't seen anything as suited for running and playing (and falling, and smacking knees) as grass. Good luck, sounds delightful!