Applying signal detection theory to single-stimulus presentations Normally, signal detection theory (SDT) is applied to data that involves giving two-alternative forced choice (2AFC) responses to pairs of stimuli, for instance saying whether stimulus B was the same or different from stimulus A. 

I am wondering if SDT can also be applied to such judgements being made of single-stimulus presentations, particularly ones for which the "correct" answer is not necessarily objectively-determined, such as, for instance, seeing a visual stimulus and deciding whether it belongs to category A or category B.
There are a huge number of paradigms that SDT can be applied to. The simplest is probably the so-called yes/no paradigm. You present a single stimulus (typically noise alone or signal plus noise) and ask was the signal present. The subject if forced to respond with either yes or no. This type of paradigm typically leads to a response bias. In a 2-interval, 2-alterantive, Forced-Choice (2I-2AFC) paradigm both a noise alone stimulus and a signal plus noise stimulus are presented in random order. Subjects are then forced to to respond if the signal was in the first or second interval. The randomization tends to reduce (but not necessarily eliminate) the response bias linked to the stimulus (subjects may respond first more often than second, but they will be randomly allocated to noise and signal plus noise trials). One can both increase the number of intervals and/or the number of responses. For example a 1I-3AFC paradigm might involve responses of red, green, or blue. A typical 3I-AFC is the AXB paradigm. Confidence rating paradigms also fall within the scope of SDT.

Without objective definitions of the "noise" and "signal plus noise" categories, it is not possible to define a hit or a correct rejection. As hits and correct rejections are the basis of SDT, it is not clear if SDT will be useful.