Causal discovery from observational data holds great promise, but existing methods rely on strong assumptions about the underlying causal structure, often requiring full observability of all relevant variables. We tackle these challenges by leveraging the score function $\nabla \log p(X)$ of observed variables for causal discovery and propose the following contributions. First, we fine-tune the existing identifiability results with the score on additive noise models, showing that their assumption of nonlinearity of the causal mechanisms is not necessary. Second, we establish conditions for inferring causal relations from the score even in the presence of hidden variables; this result is two-faced: we demonstrate the score's potential to infer the equivalence class of causal graphs with hidden variables (while previous results are restricted to the fully observable setting), and we provide sufficient conditions for identifying direct causes in latent variable models. Building on these insights, we propose a flexible algorithm suited for causal discovery on linear, nonlinear, and latent variable models, which we empirically validate.