Concerns over the supply of highly-skilled (HS) science, technology, engineering and maths (STEM) workers are well established and have been a feature of policy discourse in the UK for more than 50 years. Since the 2016 referendum on leaving the European Union, these concerns have been exacerbated by uncertainty about the movement of labour between UK and Europe. More recently, the COVID-19 pandemic has highlighted the importance of STEM skills in a wide range of areas. However, despite continued government investment in initiatives to address these concerns, the evidence base for shortages is neither well-established nor compatible with economic theories of labour supply. In order to fill a gap in the current evidence, we report on a unique analysis following the career destinations of STEM graduates from the 1970 British Cohort Study. While only a minority of STEM graduates ever work in highly-skilled STEM jobs, we identified three particular characteristics of the STEM labour market that may present challenges for employers: STEM employment appears to be predicated on early entry to the sector; a large proportion of STEM graduates are likely to never work in the sector; and there may be more movement out of HS STEM positions by older workers than in other sectors.