Segmenting curvilinear structures like blood vessels and roads poses significant challenges due to their intricate geometry and weak signals. To expedite large-scale annotation, it is essential to adopt semi-automatic methods such as proofreading by human experts. In this abstract, we focus on estimating uncertainty for such tasks, so that highly uncertain, and thus error-prone structures can be identified for human annotators to verify. Unlike prior work that generates pixel-wise uncertainty maps, we believe it is essential to measure uncertainty in the units of topological structures, e.g., small pieces of connections and branches. To realize this, we employ tools from topological data analysis, specifically discrete Morse theory (DMT), to first extract the structures and then reason about their uncertainties. On multiple 2D and 3D datasets, our methodology generates superior structure-wise uncertainty maps compared to existing models.