This study investigates the potential of language models to improve the classification of labor market information by linking job vacancy texts to two major European frameworks: the European Skills, Competences, Qualifications and Occupations (ESCO) taxonomy and the European Qualifications Framework (EQF). We examine and compare two prominent methodologies from the literature: Sentence Linking and Entity Linking. In support of ongoing research, we release an open-source tool, incorporating these two methodologies, designed to facilitate further work on labor classification and employment discourse. To move beyond surface-level skill extraction, we introduce two annotated datasets specifically aimed at evaluating how occupations and qualifications are represented within job vacancy texts. Additionally, we examine different ways to utilize generative large language models for this task. Our findings contribute to advancing the state of the art in job entity extraction and offer computational infrastructure for examining work, skills, and labor market narratives in a digitally mediated economy. Our code is made publicly available: https://github.com/tabiya-tech/tabiya-livelihoods-classifier