We study phase portraits at singular points of vector fields of the special type where all components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. Also we assume some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three applications in differential geometry: singularities of geodesic flows in various singular metrics on two-dimensional manifolds.