A configuration of the plane is a subset (both finite or infinite) A of points in \(\mathbb {Z}^2\) . Provided a finite window probe W, i.e., a finite set of positions of \(\mathbb {Z}^2\) , we shift W in all possible positions on \(\mathbb {Z}^2\) and count the points of A that fit in its positions. In case all the values are equal, say h, then we say that A is h-homogeneous w.r.t. W. When W has a tile shape, i.e., a shape that can be used to tile the plane by translation, h-homogeneous configurations related to W can be decomposed into disjoint, 1-homogeneous ones, in some cases. Very few is known about the properties that a tile has to satisfy in order to allow such a decomposition. In this paper, we shed some light on the topic, by investigating the decomposability of a class of windows, single squares, i.e., those tiles that can be surrounded in one only possible way by four copies of themselves in order to produce a tiling of the plane, that constitute one of the simplest classes of tiles. We also introduce other classes of tiles that allow homogeneous, non-decomposable planar configurations. Finally, we define the notion of reducible configuration, that concerns the possibility of decomposing a configuration into smaller, non 1-homogeneous ones, moving toward the classification of tiles w.r.t. the decomposability property.