BibTex - Show ISBN number?

This is very easy with biblatex (at least as long one takes care of closing commas in .bib files):

\documentclass{article}

\usepackage{biblatex}

\usepackage{filecontents}

\begin{filecontents}{\jobname.bib}
@book{QueueSystems,
    author    = "Leonard Kleinrock",
    maintitle     = "Queueing Systems",
    volume     = "1",     
    title     = "Theory",
    publisher = "Wiley-Interscience",
    year      = "1975",
    ISBN      = "0471491101",
}
\end{filecontents}

\addbibresource{\jobname.bib}

\nocite{*}

\begin{document}

\printbibliography

\end{document}

For more information about "first steps" with biblatex see What to do to switch to biblatex?

You write

Is it possible to show the ISBN/ISSN fields of books with BibTeX? ... I've tried several different styles with no success...

The answer is "Yes." However, in order to achieve your goal you need to use a bibliography style file that's of more recent vintage than plain.bst. The plain style file, which is more than twenty years old by now, is not set to recognize fields such as url, isbn, issn, and doi. Fortunately, a lot of the newer style files -- such as plainnat, unsrtnat, and IEEEtran -- do know how to process these fields.

For instance, with the plainnat style file and the natbib package loaded, the reference in question would be printed as follows:

Under the conditions

• You don't want to switch to biblatex,
• You don't want to change your .bst,

you can always put the ISBN (or anything else) in a note:

@book{lic,
  [...],
  note={{ISBN:} 978-91-637-4473-0}
}