The neighbors thought it was spiderwort (aka 'wandering jew'), but it doesn't really look like the photos of spiderwort I've found online. Although there seem to be a lot of species of that.
This is tree-of-heaven shoots; looks like what I remember from last year. That's not the same as the above at all, right?
Actually now, looking at those images side by side, I've calmed down a little, because I'm sure they're not the same. The lower has a shooting upright habit; the upper ones are creeping, for one thing. And the leaf shape and patterning is different. But I'd still like to know what the top-pictured plant is, as it's all over my yard. Is it invasive? Do I need to worry about it, or will it just work nicely in my to-be-mulched perennial beds, mixing happily with other plants?