Both of the complete graphs from which the Moser spindle is formed require four colors, and the Hajós construction preserves this property.

By the de Bruijn–Erdős theorem (with the assumption that the axiom of choice is true), the chromatic number of the plane is the same as the largest chromatic number of any of its finite subgraphs; until the discovery of a family of 5-chromatic unit distance graphs in 2018, no subgraph of the infinite unit distance graph had been found that requires a larger number of colors than the Moser spindle.

[2] The Moser spindle is a planar graph, meaning that it can be drawn without crossings in the plane.

The Moser spindle is also a Laman graph, meaning that it forms a minimally rigid system when embedded in the plane.

[8] As a planar Laman graph, it is the graph of a pointed pseudotriangulation, meaning that it can be embedded in the plane in such a way that the unbounded face is the convex hull of the embedding and every bounded face is a pseudotriangle with only three convex vertices.

These two properties of the Moser spindle were used by Horvat, Kratochvíl & Pisanski (2011) to show the NP-hardness of testing whether a given graph has a two-dimensional unit distance representation; the proof uses a reduction from 3SAT in which the Moser spindle is used as the central truth-setting gadget in the reduction.

[8] The Moser spindle can also be used to prove a result in Euclidean Ramsey theory: if T is any triangle in the plane, and the points of the plane are two-colored black and white, then there is either a black translate of T or a pair of white points at unit distance from each other.