One way is to use desingularization of such point and to study the local stability near the couple of more regular points or to study behavior asymptotically
Another way would be to use the center manifold theorem. To simplify, we assume 0 is the only imaginary root of the characteristic equation (other cases can be dealt with in a similar way but the calculations can be too difficult) and that all the other eigenvalues have strictly negative real part (otherwise the fixed point would be unstable).
The center manifold theorem consists of reducing the local stability analysis of any DDE to that of a system ODEs (with dimension being equal to the multiplicity of zero as a root of your characteristic equation).
The calculations of the center manifold can be tedious but doable. The resulting system ODEs may be difficult to analyse though.