Based on the principle of force equilibrium, this paper employs the Navier-Stokes equations in cylindrical coordinates (r, θ, z)to derive the three-dimensional spiral vortex velocity fields of mid-latitude cyclones, typhoons, and tornadoes. Singular points (critical points or equilibrium states) where the three-dimensional velocity of the aforementioned three types of vortices is zero are determined, all of which are saddle-focus points. Mid-latitude cyclones feature two singular points: one at the surface and the other at the top of the troposphere. Tornadoes have a single singular point at the base of the supercell storm cloud. Typhoons also exhibits two singular points, one located at the eyewall surface and the other at the center of the eyewall on the surface. And from the velocity field, the horizontal divergence (D), vertical vorticity (ωz), and the vortex shape surface are obtained, which are consistent with observational data. We found that the formation mechanisms of the three spiral vortices are different. Specifically, mid-latitude cyclones are caused by horizontal convergence due to surface friction crossing isobars, tornadoes are caused by vortex tube stretching due to strong horizontal convergence at the bottom of storm clouds, and typhoons are caused by the rising motion of the whole cumulus cloud belt over the ocean, with divergence and convergence unchanged with height.