Summary: In this paper, we analyze ends of zero mean curvature surfaces of mixed (or non-mixed) type in the Lorentzian 3-space \(\mathbf{R}^{2,1} \). Among these, we show that spacelike or timelike planar ends are \(C^{\infty}\) in the compactification \(\hat{L}\) of \(\mathbf{R}^{2,1}\) as in the case of minimal surfaces in the Euclidean 3-space \(\mathbf{R}^3\). On the other hand, lightlike planar ends are not \(C^{\infty} \). Each lightlike planar end of a mixed type surface has the following additional parts: the converging part (a lightlike line in \(\mathbf{R}^{2,1} )\), the diverging part (the set of the points in \(\hat{L} \setminus \mathbf{R}^{2,1}\) corresponding to zero-divisors), and the border of these two parts. We show that such an end is \(C^{\infty}\) on the first two parts almost everywhere, while there appears an isolated singularity in the form of \((x^3, x\tau + \text{``higher order terms''}, \tau)\) on the border. We also show that conelike singularities of mixed type appear on the lightlike lines in special cases.