The aging global population has led to an increased prevalence of spinal pathologies, making the accurate analysis of lumbar spine Magnetic Resonance Imaging (MRI) a critical clinical task. Deep learning-based segmentation of vertebrae and intervertebral discs offers a promising avenue for automating and improving this analysis. This paper presents a comprehensive empirical study comparing the performance of five state-of-the-art deep neural network architectures (U-Net 3D, V-Net, UNETR, Swin UNETR, and U-Net with ResNet-50) in combination with four widely-used loss functions (Dice, Generalized Dice, Dice + CE, and Dice + Focal). We evaluate these 20 configurations on the public SPIDER dataset across two distinct and clinically relevant tasks: semantic segmentation (classifying voxels as vertebra, disc, or background) and instance segmentation (additionally identifying individual anatomical structures). Our findings indicate that while most models perform well on semantic segmentation, with V-Net and Swin UNETR showing superior performance, instance segmentation proves to be a significantly more challenging task that better differentiates model capabilities. In this more complex scenario, V-Net and Swin UNETR again demonstrate the most robust and accurate results. This detailed analysis of results and trade-offs provides a valuable guide for the selection of appropriate models and loss functions for lumbar spine segmentation, clarifying the strengths and weaknesses of different approaches for this important clinical problem.