Summary: Let \(G_1\), \(G_2\) be two simple connected graphs. The partially subdivision neighborhood corona of \(G_1\) and \(G_2\), denoted by \(G_1\bar\star G_2\), is obtained by taking one copy of \(G_1\) and \(| V(G_1)|\) copies of \(G_2\), and joining the neighbors of the \(i\)-th vertex of \(G_1\) to every vertex in the \(i\)-th copy of \(G_2\), then inserting a new vertex into every edge of \(G_1\). In this paper, we determine the adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum of \(G_1\bar\star G_2\) in terms of those of two factor graphs \(G_1\) and \(G_2\). In addition, as many applications of these results, we consider constructing infinite pairs of adjacency cospectral, Laplacian cospectral and signless Laplacian cospectral graphs. Moreover, we compute the number of spanning trees of \(G_1\bar\star G_2\) in terms of the Laplacian spectra of two factor graphs \(G_1\) and \(G_2\).