Substrates, and layered media in general, are ubiquitous and affect the properties of any object in their vicinity. However, their influence is, in an arbitrary framework, challenging to quantify analytically, especially for large arrays which additionally escape explicit numerical treatment due to the computational burden. In this work, we utilize a versatile T-matrix-based framework to generalize the coupled multipole model toward arbitrarily high multipole orders and substrate-supported arrays. We then employ it to study substrate-supported random/amorphous arrays of high index dielectric nanoparticles which are of wide interest due to relatively low losses and a highly tunable optical response, making them promising elements for nanophotonic devices. We discuss how multipole coupling rules evolve in the presence of a substrate in amorphous arrays for three interaction mechanisms: direct coupling between particles, substrate-mediated interparticle coupling, and substrate-mediated self-coupling. We show how the interplay of array density, distance from the substrate, and the latter's refractive index determine the optical response of an array. As an example, we use this framework to analyze refractometric sensing with substrate-supported arrays and demonstrate that the substrate plays a crucial role in determining the array sensitivity.