Target detection, tracking and classification are three essential and closely coupled subjects for most surveillance systems. In the finite set statistics (FISST) framework, this paper presents a Bayesian and recursive solution to joint detection, tracking and classification (JDTC) of a manoeuvring target in a cluttered environment, which is inspired by previous work on joint target tracking and classification in the classical Bayesian filter framework. The derived JDTC algorithm exploits the dependence of target state on target class by using class-dependent dynamical model sets. The relative merits of this JDTC algorithm are demonstrated via a two-dimensional example using a sequential Monte Carlo implementation. It is shown that handling those three closely coupled subjects jointly can achieve comparable detection and tracking performance to that of the exact filter in the FISST framework with a prior known class. The classification results are consistent with the previous work.