Spatial filtering with a desired directivity pattern is advantageous for many audio applications. In this work, we propose neural directional filtering with user-defined directivity patterns (UNDF), which enables spatial filtering based on directivity patterns that users can define during inference. To achieve this, we propose a DNN architecture that integrates feature-wise linear modulation (FiLM), allowing user-defined patterns to serve as conditioning inputs. Through analysis, we demonstrate that the FiLM-based architecture enables the UNDF to generalize to unseen user-defined patterns during interference with higher directivities, scaling variations, and different steering directions. Furthermore, we progressively refine training strategies to enhance pattern approximation and enable UNDF to approximate irregular shapes. Lastly, experimental comparisons show that UNDF outperforms conventional methods.