There is a theorem by Jean Taylor that says that an almost minimal surface in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times a line and all the faces you can make from the center of a tetrahedron and its vertices.

Is there an analogous of this for minimal surfaces (mean curvature = 0)? I know that minimal surfaces are smooth but, are there examples where they kind of have the $Y$ or the tetrahedron singularity?. It is easy to see in experiments that in real soap bubbles this singularities actually appears.

Thanks, Mario