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In this work, we study the possibility of finite-time future cosmological singularities appearing in $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the stress-energy tensor. We present the theory in both the geometrical and the dynamically equivalent scalar-tensor representation and obtain the respective equations of motion. In a background Friedmann-Lemaître-Robertson-Walker (FLRW) universe with an arbitrary curvature and for a generic $C^\infty$ function $f(R,T)$, we prove that the conservation of the stress-energy tensor prevents the appearance of sudden singularities in the cosmological context at any order in the time-derivatives of the scale factor. However, if this assumption is dropped, the theory allows for sudden singularities to appear at the level of the third time-derivative of the scale factor $a(t)$, which are compensated by divergences in either the first time-derivatives of the energy density $ρ(t)$ or the isotropic pressure $p(t)$. For these cases, we introduce a cosmological model featuring a sudden singularity that is consistent with the current measurements for the cosmological parameters, namely, the Hubble constant, deceleration parameter, and age of the universe, and provide predictions for the still unmeasured jerk and snap parameters. Finally, we analyse the constraints on a particular model of the function $f(R,T)$ that guarantees that the system evolves in a direction favorable to the energy conditions at the divergence time.