For Bayesian D-optimal design, we define a singular prior distribution for the model parameters as a prior distribution such that the determinant of the Fisher information matrix has a prior geometric mean of zero for all designs. For such a prior distribution, the Bayesian D-optimality criterion fails to select a design. For the exponential decay model, we characterize singularity of the prior distribution in terms of the expectations of a few elementary transformations of the parameter. For a compartmental model and multi-parameter logistic regression, we establish sufficient conditions for singularity of a prior distribution. For logistic regression we also obtain sufficient conditions for non-singularity. In the existing literature, weakly informative prior distributions are commonly recommended as a default choice for inference in logistic regression. Our results show that some of the recommended prior distributions are singular, and hence should not be used for Bayesian D-optimal design. Additionally, methods are developed to derive and assess Bayesian D-efficient designs for logistic regression when numerical evaluation of the objective function fails due to ill-conditioning.