This article examines whether privileged social groups have managed to monopolize access to sought-after higher education (HE) programmes in a Scandinavian, universalist welfare state (Denmark) during a period of massive educational expansion (1984-2010). I apply an effectively maintained inequality (EMI) framework in which I (i) explicitly control for periods of educational expansion, (ii) disaggregate qualitatively different HE programmes, and (iii) profit from a novel way of constructing social origin, using relative measures of parental education and income. I find that probabilities of attending HE increases for almost all groups and types of programmes, and that the increase primarily occurs in periods of cohort-adjusted HE expansion. Measured by participation ratio (i.e. relative risk), neither sons nor daughters from culturally privileged backgrounds have maintained their advantage from 1984 to 2010 in access to HE in general. However, breaking down HE into four types of university programmes supports the EMI thesis as it is operationalized in this article. So, while inequality in access to HE has been reduced overall through the expansion of the HE system, this has only been achieved by channelling students of lower-educated parents to less-prestigious programmes.