The Inspectrum can refer to either the Inspector Spacetime fictional expanded universe or the fandom dedicated to the programme Inspector Spacetime.
BTV refuses to use the term "Inspectrum" in any of its publications (such as Inspector Spacetime Monthly) as that would be a tacit admission that the programme has any kind of cohesive history or that its spin-offs and licensed media share an overarching mythology with it. Instead, they refer to the Inspector's "continuing investigations" or his "further journeys", because they want people to purchase and consume new material rather than obsess with reconciling already-purchased or viewed programming.
The Inspectrum fandom, however, similarly refuses to accept the absence of an official canon and insists on rationalising all the internal contradictions into something like a coherent narrative extending from from the Big Bang to the heat death of the universe 10100 years later. The programme's many dangling plot threads, abandoned story arcs, inconsistent character histories, etc. are regarded as merely challenges to the fandom's collective creativity.
Other Fictional Universes
The most notable example is the two 1960s "Inspector Spacetime" films, Inspector Spacetime vs. the Blorgons and Blorgons—Extortion Earth 2150, which starred Christopher Lee as the title character. Despite Lee's subsequent casting as the Second Inspector on the Inspector Spacetime programme, these two films and any related stories take place in a completely separate space-time continuum. A more infamous example is the five-hour long 1973 Inspector Spacetime musical, Inspector Spacetime and the Blorgons in the Seven Clues to Armageddon, which starred and was directed by English experimental theatre actor and comedian Ken Campbell.