This is taking over our garden and lawn- mulched beds, bare spots in the lawn, between flagstones. It seems partial to shade but likes water. Plamate undulate leaves with shallow roots that form a mat. Tiny, star-shaped white flowers. The leaves are small, about the size of a fingernail. Any ideas appreciated!

Not know where you garden (state, zone, country) I would say it looks like a species of Hydrocotyle, possibly Hydrocotyle americana.

https://www.google.com/search?...

Great tip! You are right on the money- I think it's Hydrocotyle microphylla!

We are in southern NY state, just north of New York City. Thanks so much!

I am not sure H microphylla is your species. It is a native species of New Zealand

http://www.terrain.net.nz/frie...

There are quite a few species

https://en.wikipedia.org/wiki/...

Hydrocotyle microphylla is not recorded growing in the US. There are six species recorded in NY State. None seem to be a perfect match, but the three best matches I can find are Hydrocotyle americana, Hydrocotyle ranunculoides, and Hydrocotyle sibthorpioides.