The subdued wage growth observed in many countries has spurred interest in monopsony views of regional labour markets. This study measures the extent and robustness of employer power and its wage implications exploiting comprehensive matched employer-employee data. We find average (employment-weighted) Herfindhal indices of 800 to 1,100, stable over the 1986-2019 period covered, and that typically less than 9% of workers are exposed to concentration levels thought to raise market power concerns. When controlling for both worker and firm heterogeneity and instrumenting for concentration, we find that wages are negatively affected by employer concentration, with elasticities of around -1.4%. We also find that several methodological choices can change significantly both the measurement of concentration and its wage effects.