  Let $X\subset Y(1)^n$ be a subvariety defined over a number field $\mathbb F$
and let $(P_1,\ldots,P_n)\in X$ be a special point not contained in a
positive-dimensional special subvariety of $X$. We show that the if a
coordinate $P_i$ corresponds to an order not contained in a single exceptional
Siegel-Tatuzawa imaginary quadratic field $K_*$ then the associated
discriminant $|\Delta(P_i)|$ is bounded by an effective constant depending only
on $\operatorname{deg} X$ and $[{\mathbb F}:{\mathbb Q}]$. We derive analogous
effective results for the positive-dimensional maximal special subvarieties.
  From the main theorem we deduce various effective results of Andr\'e-Oort
type. In particular we define a genericity condition on the leading homogeneous
part of a polynomial, and give a fully effective Andr\'e-Oort statement for
hypersurfaces defined by polynomials satisfying this condition.
