Demographic handwriting-based classification problems, such as gender and handedness categorizations, present interesting applications in disciplines like Forensic Biometrics. This work describes an experimental study on the suitability of deep neural networks to three automatic demographic problems: gender, handedness, and combined gender-and-handedness classifications, respectively. Our research was carried out on two public handwriting databases: the IAM dataset containing English texts and the KHATT one with Arabic texts. The considered problems present a high intrinsic difficulty when extracting specific relevant features for discriminating the involved subclasses. Our solution is based on convolutional neural networks since these models had proven better capabilities to extract good features when compared to hand-crafted ones. Our work also describes the first approach to the combined gender-and-handedness prediction, which has not been addressed before by other researchers. Moreover, the proposed solutions have been designed using a unique network configuration for the three considered demographic problems, which has the advantage of simplifying the design complexity and debugging of these deep architectures when handling related handwriting problems. Finally, the comparison of achieved results to those presented in related works revealed the best average accuracy in the gender classification problem for the considered datasets.