Do employers in "non-STEM" occupations (e.g. Graphic Designers, Economists) seek to hire STEM (Science, Technology, Engineering, and Mathematics) graduates with a higher probability than non-STEM ones for knowledge and skills that they have acquired through their STEM education (e.g. "Microsoft C#", "Systems Engineering") and not simply for their problem solving and analytical abilities? This is an important question in the UK where less than half of STEM graduates work in STEM occupations and where this apparent leakage from the "STEM pipeline" is often considered as a wastage of resources. To address it, this paper goes beyond the discrete divide of occupations into STEM vs. non-STEM and measures STEM requirements at the level of jobs by examining the universe of UK online vacancy postings between 2012 and 2016. We design and evaluate machine learning algorithms that classify thousands of keywords collected from job adverts and millions of vacancies into STEM and nonSTEM. 35% of all STEM jobs belong to non-STEM occupations and 15% of all postings in non-STEM occupations are STEM. Moreover, STEM jobs are associated with higher wages within both STEM and non-STEM occupations, even after controlling for detailed occupations, education, experience requirements, employers, etc. Although our results indicate that the STEM pipeline breakdown may be less problematic than typically thought, we also find that many of the STEM requirements of "non-STEM" jobs could be acquired with STEM training that is less advanced than a full time STEM education. Hence, a more efficient way of satisfying the STEM demand in non-STEM occupations could be to teach more STEM in non-STEM disciplines. We develop a simple abstract framework to show how this education policy could help reduce STEM shortages in both STEM and non-STEM occupations.