Everyone is aware of Dijkstra's Letters to the editor: go to statement considered harmful (also here .html transcript and here .pdf) and there has been a formidable push since that time to eschew the goto statement whenever possible. While it's possible to use goto to produce unmaintainable, sprawling code, it nevertheless remains in modern programming languages. Even the advanced continuation control structure in Scheme can be described as a sophisticated goto.

What circumstances warrant the use of goto? When is it best to avoid?

As a followup question: C provides a pair of functions, setjmp and longjmp, that provide the ability to goto not just within the current stack frame but within any of the calling frames. Should these be considered as dangerous as goto? More dangerous?

Dijkstra himself regretted that title, of which he was not responsible for. At the end of EWD1308 (also here .pdf) he wrote:

Finally a short story for the record. In 1968, the Communications of the ACM published a text of mine under the title "The goto statement considered harmful", which in later years would be most frequently referenced, regrettably, however, often by authors who had seen no more of it than its title, which became a cornerstone of my fame by becoming a template: we would see all sorts of articles under the title "X considered harmful" for almost any X, including one titled "Dijkstra considered harmful". But what had happened? I had submitted a paper under the title "A case against the goto statement", which, in order to speed up its publication, the editor had changed into a "letter to the Editor", and in the process he had given it a new title of his own invention! The editor was Niklaus Wirth.

A well thought out classic paper about this topic, to be matched to that of Dijkstra, is Structured Programming with go to Statements (also here .pdf), by Donald E. Knuth. Reading both helps to reestablish context and a non-dogmatic understanding of the subject. In this paper, Dijkstra's opinion on this case is reported and is even more strong:

Donald E. Knuth: I believe that by presenting such a view I am not in fact disagreeing sharply with Dijkstra's ideas, since he recently wrote the following: "Please don't fall into the trap of believing that I am terribly dogmatical about [the go to statement]. I have the uncomfortable feeling that others are making a religion out of it, as if the conceptual problems of programming could be solved by a single trick, by a simple form of coding discipline!"