Great question! I think that what drew me to mathematics in the first place was that mathematical results rarely become useless, just less used :-)
So, like the comments, I will try to focus on topics that, while still useful, are not active areas that would sustain a PhD candidate into their career (i.e., what horses were bad to hitch your proverbial wagon to?)
First, I think it's useful to classify dormant math fields according the underlying reason they are no longer active:
Obsolete - Here I think of those poor folks who labored to produce ever more accurate numerical tables (e.g., logs, statistical distributions, nomograms) that we can now just punch into a computer and get more accurate answers.
Overhyped - Here I'd put "chaos theory" (which was so "hot" that it got mention in Jurassic Park, of all places), and "Possibility theory"...which is supposed to be an alternative to probability theory that "fixes" its flaws. When it became apparent that Chaos Theory would not "tame chaos", nor would possiblity theory reduce risk (e.g., in building a bridge), a lot of support vaporized...although there is some useful work being done in both areas to some degree, but the research groups are pretty small compared to areas like "machine learning".
Application-dependent: Unlike general mathematical theories, which are always relevant, highly applications-oriented math can actually become useless, although the lines blur between engineering and math for this area. For an example from my area: manufactured gas plants (turn coal into gas) had a lot of complex mathematical equations for predicting gas yield and such, but now that no one uses MGP's, that math is actually useless....for now.
Sociological: Here I'd place the myriad of "smart manufacturing" fads that arose after the US got surprised by the Japanese manufactring superiority in the 1980's - 1990's. Example: six-sigma, total quality management, just-in-time....all useful, but we have a glut of these, and the shelf-life is dwindling at each iteration.
That brings me to my final comment: what is "useless" today may find reapplication later in a different context - that is the beauty of mathematics research. Think of the Euler-Beta Function in String Theory, or statistical mechanics and information theory...any "dead" of mathematics area can be resurrected by a fortuitous isomorphism. This is fundamentally different from the sciences, where, for example, it is unlikely that the theory of phlogiston or the aether (as metaphysical objects...not the associated math!) will make a comeback.