Let \(X\), \(X'\) and \(Y\) be smooth algebraic (resp. holomorphic) manifolds of the same positive dimension over the ground field \(\mathbb C\). By definition, two finite regular (resp. holomorphic) mappings \(f : X \rightarrow Y\) and \(g : X'\rightarrow Y\) are equivalent if there exists a regular (resp. holomorphic) isomorphism \(\varphi : X \rightarrow X'\) such that \(f = g\, {^{_\circ}}\,\varphi\). The author proves that for every (reduced) hypersurface \(D\subset Y\) and every \(k\in \mathbb N\) there exist only a finite number of non-equivalent finite regular mappings \(f: X\rightarrow Y\) such that the discriminant of \(f\) is equal to \(D\) and \(\mu(f)=k\). He also remarks that the same result is also true in the holomorphic setting, if one additionally assumes that the fundamental group of the space \(Y \setminus D\) is finitely generated. A series of useful applications, involving a comparison between local and global settings, the case of mappings of degree two, the case of discriminants which are divisors with simple normal crossings, and close relations with some earlier results (see, e.g., [S. Lamy, Publ. Mat., Barc. 49, No. 1, 3–20 (2005; Zbl 1094.14052); Contemp. Math. 553, 15–25 (2011; Zbl 1235.14054)]), are discussed in detail.