My neighbor's landscaper planted this groundcover (in southern California) which she told me was a thyme. It spreads like wildfire and has a tiny white flower with a raised purple center, but it neither looks nor smells like any thyme I have ever seen. Any idea what it actually is? It is evergreen, with jagged edged leaves roughly half an inch long. [groundcover leaf with jagged edges
This is Phyla nodiflora, also sometimes called Lippia repens, and apparently lots of other names (matchweed, etc...). According to Native Florida Wildflowers website, it is a Florida native. No it isn't really thyme as we know it, although maybe one of its nicknames could be "thyme". Wikipedia puts it in the verbena family.
I have it in my lawn (Northern Calif.), and yes it does spread well. As you say, like wildfire, or if you prefer, it is invasive. It seems to propagate mostly by creepers that root into the ground if undisturbed for a while. In the middle of the grass it keeps low to the ground and makes the grass look a little lusher and greener, but around the edges where there is less competition it can get tallish and rangy looking, less like a ground cover and more like a scrubby weed. That said, it is pretty easy to pull out in places you don't want it. (Not sure how easy it is to completely eradicate if you decide you hate it). Once established it doesn't seem to need a lot of water to survive, but looks nicer if it gets some.
Also please note in the spring/summer (at least in my climate), it is absolutely covered with those little white flowers which attract bees like crazy! (FYI...if you happen to have serious bee-sting allergies in your family, you might want to avoid it.)