Hi Bill. It appears to be mouse ear chickweed. The photos are a little difficult to see the details of the plant growth, but that is what it appears to be. Mouse ear spreads by seed fairly close to the original plant. It also spreads a little bit by shallow root. Most likely the reason your neighbors don't have an abundance is lawn treatment and healthier turf. It is a fairly pretty "weed" in the spring and fall but will look ratty in the summer. If you aren't one for chemical treatment of your lawn *applause* then it will, most likely continue to seed itself and thrive. If you are looking to get away from lawn turf, I would suggest white clover. Clover will draw beneficial bees and insects to your landscape, flower prolifically, fixes nitrogen, stays green even when your neighbor's chemically treated lawns are dead from drought and heat, and requires minimal mowing. Just be aware that bare feet on clover means bee stings are possible! Go to a local Garden Center that carries clover seed and talk to them about what you would like to see as a lawn.

Robbie

Here is a link that might be useful: White Clover

Robbie's advice still applies, but I think the weed you are looking at is bittercress. Give it a google

Yes, it's bitter cress. You can identify it by it's seed heads...sort of threadlike, but when ripe they will shoot seeds a long way. I just pulled it from 4 garden plots out of self preservation in our Community Garden.