We consider relativistic and non-relativistic particles and strings in spaces (or spacetimes) with a degenerate metric. We show that the resulting dynamics is described by a rich structure of constraints. All constraints are classified and the dynamics depends strongly on the parity of the difference between the dimension of the space (or spacetime) and the rank of the degenerate metric. For a particular class of degenerate metrics we can use the null eigenvectors of the metric in order to obtain an isometry condition. We also give the first steps towards the quantization of the non-relativistic particle using the Senjanovic path integral quantization and the Batalin-Fradkin-Tyutin conversion method.