This paper deals with a singular integral equation \[ Sq+Tq=f,\tag{1} \] where \(q(x)\) is an unknown function, \[ Sq(x):=aq(x)+\frac{1}{\pi }\text{v.p.} \int_{-1}^{1} \frac{q(\tau)}{\tau -x} d\tau,\;Tq(x):=\int_{-1}^{1}K(x,\tau)q(\tau) d\tau. \] It is assumed that the functions \(f\) and \(K\) smoothly depend on additional parameters. The problem is to study interrelations between singularities of solutions to (1) and those of families \(f, K\) [see \textit{V. I. Arnol'd, A. N. Varchenko} and \textit{S. M. Gusejn-Zade}, Singularities of differentiable mappings. Classification of critical points, caustics and wave fronts. (Russian) Moskva: ``Nauka'' Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury (1982; Zbl 0513.58001)]. For this purpose a notion of differentiable equivalence of integral operators is introduced. For equation (1) the authors establish how do the singularity types of its solutions depend on the dimension of families \(f\) and \(K\).