This paper presents a method for detecting independently moving objects (IMOs) from a monocular camera mounted on a moving car. We use an existing state of the art monocular sparse visual odometry/SLAM framework, and specifically attack the notorious problem of identifying those IMOs which move parallel to the ego-car motion, that is, in an `epipolar-conformant way. IMO candidate patches are obtained from an existing CNN-based car instance detector. While crossing IMOs can be identified as such by epipolar consistency checks, IMOs that move parallel to the camera motion are much harder to detect as their epipolar conformity allows to misinterpret them as static objects in a wrong distance. We employ a CNN to provide an appearance-based depth estimate, and the ambiguity problem can be solved through depth verification. The obtained motion labels (IMO/static) are then propagated over time using the combination of motion cues and appearance-based information of the IMO candidate patches. We evaluate the performance of our method on the KITTI dataset.