There are many ways a PM DC motor can fail. Most of the problems occur in the armature.
When a motor goes up in smoke, generally the wire coil on one pole of the armature has shorted out. This can usually be discerned by burned insulation on most of the wires in that coil.
Sometimes a wire will break, usually near one of the commutator tabs. This normally results in no power, running hot, and sometimes the motor will then not start by itself.
Other failure modes are having a commutator segment come loose or having a brush wear through or break off (on the motors which use leaf-spring style brushes). On motors with separate brushes and springs, the brush can get so short the spring can no longer hold it against the comm; this often burns up the comm due to excessive arcing.
Mechanical problems can also occur, such as a magnet coming loose or a bushing wearing out. Any mechanical problem preventing the arm from rotating will quickly ruin it if power is not removed in short order.