Most U.S. health surveys (NHIS and its kiddo MEPS, NHANES, NSDUH) are stratified cluster surveys. The common representation of the public use data sets is a two-stage design with ~50 strata at the first stage of sampling (at which clusters are sampled), usually with two clusters per stratum, and people sampled at the second stage within clusters. This is kind of sixth grade reading level explanation of science, if you like.
Why, and how, are these surveys stratified? Well, the health professionals know that people in different settings have different health care needs and health care outcomes. Urban is different from suburban different from rural, so the level of urbanization / population density is a stratifying variable for these.
Why, and how, are these surveys clustered? Well, cluster samples are either a measure of desperation (there is no way to reach the population in other ways), or simply a way to save on costs (in face-to-face surveys, you rather want to pay interviewers to talk with people, rather that to sit in the car / on the train / walk from one interview to next... so the interviewers should have 5-10-15 minute travel than 2 hour travel between appointments). In large scale U.S. health surveys, you have bits of both: there is no central listing of all people in the country (although one can lay their hands on the list of all addresses, sort of). In international surveys like Demographic and Health Surveys , there may not be enough government data to set up data collection like it is done in the U.S.; the best you may have to deal with is administrative division into provinces, districts, and cities/towns/villages within the latter, with at best rough estimates of population sizes. So you end up sampling those districts, and those settlements within districts, and then send enumerators to count dwellings and then sample from the lists thus created.
There are of course other situations where cluster samples make perfect sense -- namely when the populations are absolutely naturally organized in hierarchical way, like school districts / schools / classes-teachers / students. Clusters are defined by the social processes, not by the statistician's pen. In many of these hierarchical population surveys, there is also interest in data at each level of hierarchy, and in multilevel modeling of mediation of student-level variable effects by the teacher or principal-level variables.
Out of the questions posed by the OP, I can only answer this (others are qualitative research questions, not quantitative research ones):
- What circumstances would lead a study designer to say "You know what? We need an additional variable to cluster sample/stratify on."
You can only stratify on a variable that is available on the sampling frame (sampling frame = list of entities that you take a sample from; this would be a list of districts in the example of the DHS surveys, or the list of all 80,000 Census tracts in the case of the United States for the large scale health surveys; this could also be an implicit list like the way to generate random phone numbers in random digit dialing, which is what is being done for BRFSS).
As far as to which variable is to cluster on, it is either the natural hierarchy, or a cost-precision tradeoff: if your interviewers have smaller area to cover, the population is likely to be somewhat more homogeneous, so you don't learn as much from the same number of observations.
P.S. The distinction between clusters and strata is something a lot of people struggle with. You are not alone.
P.P.S. Contrary to what you may have heard, including some of the posted answers, in the U.S., you cannot stratify by person's race/ethnicity, sex/gender, or age, not in the general population surveys, at least. If you have a list of hospital patients with these fields, then of course you can. But there is no general sampling frame (short of maybe the Census Bureau Master Address File) that would list person's name, person's address, and these demographic characteristics. The Nordic countries, however, have population registers where this information can be found; the conversations between Swedes and Americans at professional conferences sometimes go in parallel universes with little traction.) What does happens is that when you stratify by geography, and minorities are heavily segregated, you can select areas that are 90%+ Black/African American or 80%+ Hispanic, and that way you have a good way to predict how many people in those groups your sample will have in the end of the day.