In this paper we extend a scalar Q-convexity shape descriptor in two ways towards rotational and rotation-free multidirectional descriptors. Since the original descriptor is defined with respect to the horizontal and vertical directions, we increase its ability to represent a shape by repeatedly rotating the shape and computing the descriptor, in the first extension. However, since rotation is generally not a safe transformation in the 2D digital space, as a second approach, we develop an alternative that works directly with an arbitrary pair of orthogonal lattice directions: the shape remains fixed and a certain number of pairs of directions are employed. We provide the algorithmic details of this approach and its computational complexity. In case of both approaches, the Q-concavity values measured along different directions can be gathered into a vector multidirectional descriptor (signature). We conducted experiments on the MPEG-7 dataset with different sets of lattice directions. We conclude that even though the rotational signature is faster to compute, the rotation-free signature can ensure better classification accuracy. Depending on whether computational time or accuracy is more important in the given application, one of the two methods can be preferred to the other, and hence both of them seem to provide a viable option.