In this paper, mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations with initial singularity are investigated. By adopting the modified Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution. Furthermore, the regularity and logarithmic decay of its solution are researched. Under the singularity hypothesis, the numerical methods for the problems are then studied. The model is first transformed into its equivalent form and then the technic of exponential type meshes is utilized. The fact that the discrete coefficients of Hadamard fractional integral have several graceful properties is crucial in convergence and stability analysis. For the sake of reducing storage and computational cost, the SOE technology is exploited to the new variable [Formula presented]. On this basis, a fast compact difference scheme and a fast compact ADI method are constructed for the one- and two-dimensional problems, respectively. To illustrate the significance of studying singularity, a Crank-Nicolson difference scheme is proposed. The final result that the error in numerically approximating the solution depends on the parameter γ shows obviously how the regularity of the solution and the exponential type meshes affect the convergence order of the derived scheme. Ultimately, the numerical examples are provided to verify that the fast algorithm effectively reduces the computational cost compared to the direct method.