> What do our metropolitan areas look like—what shape, or model, best describes them—today?

I can answer this for Texas, except Houston. Houston is an outlier among all major US cities.

Large urban centers are a relative modern phenomenon in Texas beginning around the late 19th century. Cities existed in Texas before that, but they weren't what we would think of today as urban. In most of the first half of the 20th century Texas cities weren't that large and everybody lived in the city near the city center where the jobs were. Things were diverse and there weren't indicators of racial neighborhoods or stratification.

Then around the 1950s things changed and affluent white people left the cities in droves to found suburbs on the city edges. This is due to a number of factors: racial integration of the schools, growing population densities, traffic, strain on urban services. This movement away from the cities left neighborhoods lacking the prior diversity and arose racial neighborhoods. Keep in mind that racial integration started in schools with the Brown decision in 1952, but would not be reflected in much of the rest of society into the Civil Rights Act of 1964. Most unique though suburbs added larger living spaces than cities could afford, which was attractive to people who had the means to move.

Growth continued and continued. When suburbs became too dense, too noisy, or grew away from the previous affluence more suburbs were founded outside the existing suburbs and living conditions became more spacious still. The pattern is that city living would become poorer and progressively more affluent the further away you went. What's interesting though is that several of the large cities saw this early and corrected for it. Those cities still have some poor inner-city neighborhoods but have made early efforts to attract money back into the urban core. This has resulted in wealthy city centers surrounded by a ring of poverty at the edge of the old city boundaries and inner suburbs.

Another solution of the big cities is to use their voting power and tax wealth to annex land fast than people can found or grow suburbs thereby growing the space of the city to create suburbs outside the farther suburbs yet still within the city boundaries of the big city.

This continually outward expansion and large residential estates created a new problem: long commutes. Texas has always had really weak public transit. There have been numerous proposals and bond packages to fix this over the years, but they are always killed for political reasons. The airline industry is huge in Texas so that are particularly quick to kill any kind of long distance public transit. Texas has spent more money on its freeways than perhaps every other US state and more than a good portion of the remaining country combined, but the freeways are really nice. If you want to defeat public transportation with urban areas that wide and populations that large you need a really good excuse, and limiting traffic congestion is a good half answer.

With commutes getting longer and longer there is appeal to giving up the larger yards and moving back to the city. People are working longer than they used to and certainly don't want to be spending a good portion of additional time driving. The demand is high enough that some people are willing to pay substantially more for smaller spaces to be closer to work.