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A true labor shortage is when no labor is available for a task at any price. Surely this occurs all of the time if you consider markets that don't exist or could never exist, but has this ever happened otherwise?
Has there ever been a situation where a particular type of labor existed, and then dissapeared, leading to a situation where none of the labor in that sector was available at any price?
That very narrow definition of labour shortage, "when no labor is available for a task at any price", is rarely used, because it's a very rare phenomenon. Market forces kick in, raising prices sharply enough to suppress even relatively inelastic demand, long before the shortage becomes absolute. So the phrase is much more commonly used to mean that there is a temporary constriction in supply of a particular type of labour, usually leading to a large, rapid increase in its remuneration. Hence, when you hear the phrase, that's much more likely to be the intended meaning.
However, in the very narrow definition, the canonical example given is usually that of many trades in Western Europe, after outbreaks of the Black Death in the Fourteenth Century had halved the population.
However, even that example is problematic, as the evidence is that women were largely prevented from taking men's jobs, so the shortage was as much about culture as about physical limits:
Labour scarcity should have opened up opportunities for females, but the regulations only added to their geographical and occupational immobility,
A "true" labour market shortage as defined by no labour at any price has probably never occurred. Let me argue why:
Who defines the any price? There is always a price (wage) someone is willing to accept to move to a certain location, or to perform a job that does not match own preferences, or for which training is willing to be carried out in order to be able to perform that task/job. This is, there is always a non-infinite price at which you could fill a post. In the example of the Black Death, I am absolutely sure that if you were able to offer a price to Chinese workers to go to Europe they would have done so. As such, using your definition of shortage, no, it was not a shortage.
The key factor is then not the supply of workers but the demand for workers. In other words, why is not a given post filled in? It is because the employer is not willing to pay the price at which someone would fill up the position. So, returning to the Black Death example, there was "a shortage" only because employers (farm owners, mostly) were not willing to pay high enough wages to motivate enough people to migrate to Europe.
My bottom line is that your extreme definiton of labour market shortage as a "no price" market is too extreme, and a more useful, albeit subjective definition is based on a relative perspective. Nothing else than out demand and supply paradigm.
Note: in a simple demand and supply graph for the labour market, do not forget that there is somewhere to the right another line which represents the "total employment" available. Although we usually represent this as perfectly inelastic, in fact, this line is also endogenous to the wage level. That is how you can incorporate immigration to the model. Under this view, the actual "total" supply is never inelastic. As such, there is always a positive, non-infinite price at which you can command workers into a job.
