Introduction: A phenomenological approach is proposed to monitor the propagation of the first waves of the COVID-19 pandemic. Method: A large set of data collected during the first months of 2020 is compiled into a series of semi-logarithmic plots, for a selection of 32 countries from the five continents. Results: Three regimes are identified in the propagation of an epidemic wave: a pre-epidemic regime 1, an exponential-growth regime 2, and a resorption regime 3. A two-parameters scaling of the first-wave death variation reported in China is used to fit the first-wave data reported in other countries. Comparison is made between the propagation of the pandemic in different countries, which are classified into four groups, from Group A where the pandemic first waves were contained efficiently, to Group D where the pandemic first waves widely spread. All Asian countries considered here, where fast and efficient measures have been applied, are in Group A. Group D is composed of Western-European countries and the United States of America (USA), where late decisions and confused political communication (pandemic seriousness, protection masks, herd immunity, etc.) led to a large number of deaths. Discussion: The threat of resurging epidemic waves following a lift of lockdown measures is discussed. The results obtained in Asian countries from group A, as Hong Kong and South Korea, are highlighted, and the measures taken there are presented as examples that other countries may follow.