Segmenting cytoskeletal filaments in microscopy images is essential for studying their roles in cellular processes. However, this task is highly challenging due to the fine, densely packed, and intertwined nature of these structures. Imaging limitations further complicate analysis. While deep learning has advanced segmentation of large, well-defined biological structures, its performance often degrades under such adverse conditions. Additional challenges include obtaining precise annotations for curvilinear structures and managing severe class imbalance during training. We introduce a novel noise-adaptive attention mechanism that extends the Squeeze-and-Excitation (SE) module to dynamically adjust to varying noise levels. Integrated into a U-Net decoder with residual encoder blocks, this yields ASE_Res_UNet, a lightweight yet high-performance model. We also developed a synthetic dataset generation strategy that ensures accurate annotations of fine filaments in noisy images. We systematically evaluated loss functions and metrics to mitigate class imbalance, ensuring robust performance assessment. ASE_Res_UNet effectively segmented microtubules in noisy synthetic images, outperforming its ablated variants. It also demonstrated superior segmentation compared to models with alternative attention mechanisms or distinct architectures, while requiring fewer parameters, making it efficient for resource-constrained environments. Evaluation on a newly curated real microscopy dataset and a recently reannotated dataset highlighted ASE_Res_UNet's effectiveness in segmenting microtubules beyond synthetic images. For these datasets, ASE_Res_UNet was competitive with a recent synthetic data-driven approach that shares two cytoskeleton pretrained models. Importantly, ASE_Res_UNet showed strong transferability to other curvilinear structures (blood vessels and nerves) across diverse imaging conditions.