Anzahl theorems in finite singular symplectic, unitary and orthogonal geometries

Abstract Transitive sets of subspaces of a vector space over a finite field under extended symplectic, unitary and orthogonal groups are determined and their cardinals are computed. The number of subspaces in a transitive set contained in a given subspace of the ordinary symplectic, unitary or orthogonal geometries is also computed. These numbers have obvious applications in block designs.