If we could all see weeds as "wildflowers" - whether they are or not - how much easier it would be! There are some perennial weeds that I just hate the sight of, like henbit. Why this one in particular bugs me, I'm not sure, as you wouldn't necessarily consider it ugly, but I do take its appearance as nearly a personal affront. On the other hand, there are some perennail groundcover weeds I can easily live with. One is Duchesnea indica, known as Indian mock strawberry.
My first year here, I noticed this rather attractive plant, looking like a wild strawberry. Just a few inches high, bright green leaves, in threes, spreading by runners; small yellow flowers and eventually little red fruits. Looked like a strawberry plant, acted like a strawberry plant, must be - ? Called the local extension office but our officer there had no idea. So I looked online and found it. NO relation to strawberries (Fragaria) at all. And I had noted that the little fruits were actually not so interesting: no scent, small, a little grainy. Reportedly, no scent, no taste, but innocuous, nothing toxic here. So I decided this is one I can live with. I find it quite an attractive groundcover, really, and it's easy to pull up when it starts crowding something I've planted. There are some bare areas under the crepe myrtles that have been hard to deal with, especially since I found that voles will devour any hosta I plant. So this is my solution: love the one you're with, so to speak.