A singular modulus is the j-invariant of an elliptic curve with complex multiplication; as such the arithmetic and algebraic properties of these numbers are of great interest. In particular, there are important results concerning the behavior of differences of singular moduli, and also about the multiplicative dependencies that can arise among singular moduli. In joint work with Vahagn Aslanyan and Guy Fowler we show that for every positive integer n there are a finite set S and finitely many algebraic curves \(T_1,...,T_k\) with the following property: if \((x_1,...,x_n,y)\) is a tuple of pairwise distinct singular moduli so that the differences \( (x_1-y),...,(x_n-y)\) are multiplicatively dependent, then \( (x_1,..., x_n, y)\) belongs either to \(S\) or to one of the curves \(T_1,...,T_k\).