  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-Lema\^{i}tre-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 $\rho(t)$ 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.
