  A timelike minimal surface in Minkowski 3-space is a surface whose induced
metric is Lorentzian and with vanishing mean curvature. Such surfaces have many
kinds of singularities. In this paper, we prove existence and non-existence
theorems of singularities of timelike minimal surfaces, and show that various
diffeomorphism types of singularities that do not appear on these Riemannian
counter parts, such as minimal surfaces in Euclidean space and maximal surfaces
in Minkowski space, appear on timelike minimal surfaces. We also give criteria
for cuspidal butterfly, cuspidal $S_1$ singularity, $(2,5)$-cuspidal edge,
cuspidal beaks and $D_4$ singularity of timelike minimal surfaces. Finally,
duality and invariance theorems for these singularities and examples are given.
