We study metrizable projective structures near non-linearizable singularities of projective vector fields. We prove connected 3-dimensional Riemannian manifolds and closed connected pseudo-Riemannian manifolds admitting a projective vector field with a non-linearizable singularity are projectively flat. We also show that a 3-dimensional Lorentzian metric is projectively flat on a cone with its vertex at non-linearizable singularities of projective vector fields.