Deviant case sampling is one form of "useful" nonprobability sampling. It is useful in the sense that it allows researchers to observe rare or special groups from the population of interest. A deviant case sample, by definition, contains individual data points that have statistically significantly large values. An example of a deviant case sample would be data from swamps and deserts when comparing landscapes on water content.
A maximum variation sample is one that attempts to obtain a large amount of variation on the parameter of interest. In random sampling, there may be groups that do not find their way into the sample because of their relatively small size or rarity. Maximum variation sampling gives a "fairer" sample in that researchers endeavor to include groups that would normally not be included in the study. For example, when researching world work and income trends, researchers may want to purposefully include residents of small countries such as Singapore and Luxemburg, since individuals from these countries are not likely to be randomly chosen by the traditional random sampling method.
In situations of opinion or information-gathering, researchers may pursue samples filled with known experts instead of laypersons. This type of sample is usually not representative of the public or issue at hand, but can often provide information of higher quality. For example, singing competitions and record companies frequently use ratings from panels of experts on music instead of the average person, even though it is the average person who will be buying the CDs.
This form of sampling is simply "sampling with purpose." Researchers using a purposive sample know the groups in the population they want information from, even though these groups cannot represent the population on a whole. This form of sampling is direct and convenient and is often used in situations when representation is not an important (i.e., the researchers do not intend to make many general inferences from the sample).