The authors analyze the convergence properties of certain polynomial spline collocation solutions \(u\) for second-kind Volterra integral equations of the form \[ y(t)= \int^ t_ 0 p(t,s)k(t,s,y(s))ds+ g(t),\quad t\in I:= [0,T],\tag{1} \] where \(k\) is smooth and \(p\) is unbounded but integrable; typically, \(p(t,s)= (t- s)^{-\alpha}\), \(\alpha\in (0,1)\). The main part of the paper deals with the case where the analytical solution \(y\) of (1) is in \(C^ m(I)\) \((m\geq 1)\): if \(u\) lies in the space \(S^{(-1)}_{m-1}(Z_ N)\) (discontinuous splines of degree \(m-1\)) or in \(S^{(0)}_{m-1}(Z_ N)\) (continuous splines of degree \(m-1\)), then on uniform meshes \(Z_ N\) the collocation error behaves like \(\| y- u\|_ \infty\leq Ch^ m\) as \(h\) tends to zero. The arguments used in the proof of this result are slight generalizations of those employed by the reviewer and S. P. Nørsett [Numer. Math. 36, 347-358 (1981; Zbl 0451.65100)] and J. Abdalkhani [Congr. Numerantium 42, 87-100 (1984; Zbl 0539.65093)].
In general, however, one has \(y\in C^ m(0,T]\cap C[0,T]\), with \(y'\) unbounded at \(t= 0\). It is shown that for \(p(t,s)= (t- s)^{-1/2}\) and smooth \(g\) and \(k\), the transformation \(t= v^ 2 T\) in (1) leads to an integral equation whose solution is smooth so that the above result on the optimal convergence rate holds. A numerical example is used to confirm the theory and to compare the performance of the method with the one of J. Norbury and A. M. Stuart [Proc. R. Soc. Edinb. Sect. A 106, 361-373 and 375-384 (1987; Zbl 0639.65075 and Zbl 0639.65076)] and two product integration methods.