A monoid surface is a surface of degree d which has a singular point of multiplicity d-1. Any monoid surface admits a rational parameterization, hence is of potential interest in computer aided geometric design. The possible real forms of the singularities on a monoid surface are determined. These results are applied to the classification of quartic monoid surfaces and a study of a stratification of the parameter space of these surfaces. A part of this work is joint with M. Løberg and P.H. Johansen, the other part is due to P.H. Johansen.