On a regular grid, the analysis of digital straight lines (DSL for short) has been intensively studied for nearly half a century. In this article, we attend to multi-scale discrete geometry. More precisely, we are interested in defining geometrical properties on heterogeneous grids that are mappings of the Euclidean plane with different sized isothetic squares. In some applications, such a heterogeneous grid can be a hierarchical subdivision of a regular unit grid. First of all, we define the objects in such a geometry (heterogeneous digital objects, arcs, curves...). Based on these definitions, we characterize the DSL on such grids and then, we develop the algorithms to recognize segments and to decompose a curve into maximal pieces of DSL. Finally, both algorithms are illustrated and practical examples that have motivated this research are given.