  We show by phase space analysis that there are exactly 17 possible
qualitative behaviors for a rotational surface in $\mathbb{R}^3$ that satisfies
an arbitrary elliptic Weingarten equation $W(\kappa_1,\kappa_2)=0$, and study
the singularities of such examples. As global applications of this
classification, we prove a sharp halfspace theorem for general elliptic
Weingarten equations of finite order, and a classification of peaked elliptic
Weingarten spheres with at most two singularities. In the case that $W$ is not
elliptic, we give a negative answer to a question by Yau regarding the
uniqueness of rotational ellipsoids.
