Given a surface $S$ and an integer $r\,{\ge}\,1$, there is a variety $X_{r-1}$ parametrizing all clusters of $r$ proper and infinitely near points of $S$. We study the geometry of the varieties $X_r$, showing that for every Enriques diagram ${\bf D}$ of $r$ vertices the subset ${\mathit{Cl}}({\bf D})\,{\subset}\,X_{r-1}$ of the clusters with Enriques diagram ${\bf D}$ is locally closed. We study also the relative positions of the subvarieties ${\mathit{Cl}}({\bf D})$, showing that they do not form a stratification and giving criteria for adjacencies between them.